diff --git a/README.md b/README.md
index 5ac82978..a3bb4b12 100644
--- a/README.md
+++ b/README.md
@@ -37,7 +37,7 @@ npm run build        # writes dist/ and regenerates index.html; the folder is wh
 The browser suite and the three example games need a machine with a GPU:
 
 ```bash
-node tools/seed-e2e.mjs && node tools/e2e.mjs      # the editor, 742 checks (GM_STEPS=20 for one step)
+node tools/seed-e2e.mjs && node tools/e2e.mjs      # the editor, 758 checks (GM_STEPS=20 for one step)
 node tools/seed-jumprun.mjs && node tools/smoke-jumprun.mjs
 node tools/seed-tetris.mjs && node tools/smoke-tetris.mjs
 node tools/seed-actor.mjs && node tools/smoke-actor.mjs
diff --git a/src/core/commands/containerCommands.ts b/src/core/commands/containerCommands.ts
index 4bae243c..777cc3c9 100644
--- a/src/core/commands/containerCommands.ts
+++ b/src/core/commands/containerCommands.ts
@@ -1,18 +1,18 @@
 import { newId } from '../ids';
 import type { ContainerAsset } from '../model/Asset';
-import {
-  findObject,
-  localPositionFor,
-  removeSubtrees,
-  subtreeIds,
-  subtreeRoots,
-  treeOrder,
-  worldPosition,
-} from '../model/hierarchy';
+import { findObject, removeSubtrees, subtreeIds, subtreeRoots, treeOrder } from '../model/hierarchy';
 import { type InstanceEdit, isLiveInstance, materializeInstance } from '../model/instance';
+import { apply, invert, multiply, translation } from '../model/matrix';
 import type { Position } from '../model/Properties';
-import { findObjectHolder, type Project } from '../model/Project';
+import { findAsset, findObjectHolder, type Project } from '../model/Project';
 import { createSceneObject, type SceneObject } from '../model/SceneObject';
+import {
+  carryTransform,
+  localPositionFor,
+  parentTransform,
+  worldPosition,
+  worldTransform,
+} from '../properties/transform';
 import { removeById, type Removed, restoreAt } from '../util/list';
 import { buildAsset } from './assetCommands';
 import { type Command, CommandError } from './Command';
@@ -42,7 +42,7 @@ export class CreateContainerCommand implements Command {
     const selected = ids.map((id) => requireStructuralObject(project, holder, id));
     if (selected.length === 0) throw new CommandError('Select at least one object to create a container');
 
-    const worlds = selected.map((o) => worldPosition(holder, o));
+    const worlds = selected.map((o) => worldPosition(project, holder, o));
     const centre: Position = {
       x: Math.round((Math.min(...worlds.map((p) => p.x)) + Math.max(...worlds.map((p) => p.x))) / 2),
       y: Math.round((Math.min(...worlds.map((p) => p.y)) + Math.max(...worlds.map((p) => p.y))) / 2),
@@ -68,31 +68,41 @@ export class CreateContainerCommand implements Command {
       nextAssetName(project, 'Container'),
       this.assetId,
     ) as ContainerAsset;
+    const containerLocal = localPositionFor(project, holder, containerParentId, centre);
+    // The template is the container's frame: a root keeps its place and look inside it, which
+    // under a turned or sized parent means transform rows of its own (see `carryTransform`).
+    const containerWorld = multiply(
+      parentTransform(project, holder, containerParentId).m,
+      translation(containerLocal.x, containerLocal.y),
+    );
     const templateIds = this.templateIds;
     for (const object of members) if (!templateIds.has(object.id)) templateIds.set(object.id, newId());
     for (const object of members) {
       const isRoot = roots.has(object.id);
-      const world = worldPosition(holder, object);
-      asset.objects.push({
+      const world = worldTransform(project, holder, object);
+      const inside = apply(invert(containerWorld), { x: world.m.tx, y: world.m.ty });
+      const template: SceneObject = {
         ...object,
         id: templateIds.get(object.id)!,
-        position: isRoot
-          ? { x: world.x - centre.x, y: world.y - centre.y, z: world.z }
-          : { ...object.position },
+        position: isRoot ? { x: inside.x, y: inside.y, z: world.z } : { ...object.position },
         // A view-only offset belongs to the placed object, not to the template.
         editorOffset: { x: 0, y: 0 },
         overrides: { ...object.overrides },
         parentId: isRoot ? '' : (templateIds.get(object.parentId) ?? ''),
         instanceEdits: { ...object.instanceEdits },
-      });
+      };
+      const objectAsset = findAsset(project, object.assetId);
+      if (isRoot && objectAsset)
+        carryTransform(
+          objectAsset,
+          template,
+          parentTransform(project, holder, object.parentId).m,
+          containerWorld,
+        );
+      asset.objects.push(template);
     }
 
-    const container = createSceneObject(
-      this.objectId,
-      this.assetId,
-      localPositionFor(holder, containerParentId, centre),
-      containerParentId,
-    );
+    const container = createSceneObject(this.objectId, this.assetId, containerLocal, containerParentId);
     this.removed = removeSubtrees(holder, [...roots]);
     project.assets.push(asset);
     holder.objects.push(container);
diff --git a/src/core/commands/sceneObjectCommands.ts b/src/core/commands/sceneObjectCommands.ts
index e949fee2..89a9bbbb 100644
--- a/src/core/commands/sceneObjectCommands.ts
+++ b/src/core/commands/sceneObjectCommands.ts
@@ -1,5 +1,5 @@
 import { newId } from '../ids';
-import { isSelfOrAncestor, localPositionFor, removeSubtrees, worldPosition } from '../model/hierarchy';
+import { isSelfOrAncestor, removeSubtrees } from '../model/hierarchy';
 import {
   captureObject,
   isLiveInstance,
@@ -8,10 +8,23 @@ import {
   restoreObject,
 } from '../model/instance';
 import { placementRejection } from '../model/placement';
-import { type Position, type PropertyValue, toPosition, type Vec2 } from '../model/Properties';
+import {
+  type Position,
+  type PropertyBag,
+  type PropertyValue,
+  toPosition,
+  type Vec2,
+} from '../model/Properties';
 import type { Project } from '../model/Project';
 import { createSceneObject, type SceneObject } from '../model/SceneObject';
 import { coerceValue } from '../properties/PropertySchema';
+import {
+  carryTransform,
+  localPositionFor,
+  parentTransform,
+  positionOf,
+  worldTransform,
+} from '../properties/transform';
 import { type Removed, restoreAt } from '../util/list';
 import { type Command, CommandError } from './Command';
 import {
@@ -75,7 +88,7 @@ export class PlaceAssetCommand implements Command {
     const rejection = placementRejection(project, holder, asset);
     if (rejection) throw new CommandError(rejection);
     if (this.parentId) requireStructuralObject(project, holder, this.parentId);
-    const local = localPositionFor(holder, this.parentId, toPosition(this.position));
+    const local = localPositionFor(project, holder, this.parentId, toPosition(this.position));
     holder.objects.push(createSceneObject(this.objectId, this.assetId, local, this.parentId));
   }
 
@@ -256,9 +269,10 @@ export class ReorderObjectCommand implements Command {
  */
 export class SetParentCommand implements Command {
   readonly label = 'Set parent';
-  private previous: { parentId: string; position: Position } = {
+  private previous: { parentId: string; position: Position; overrides: PropertyBag } = {
     parentId: '',
     position: { x: 0, y: 0, z: 0 },
+    overrides: {},
   };
 
   constructor(
@@ -277,15 +291,29 @@ export class SetParentCommand implements Command {
       if (isSelfOrAncestor(holder, this.objectId, this.parentId))
         throw new CommandError('An object cannot be parented to one of its own children');
     }
-    this.previous = { parentId: object.parentId, position: { ...object.position } };
-    const world = worldPosition(holder, object);
+    this.previous = {
+      parentId: object.parentId,
+      position: { ...object.position },
+      overrides: { ...object.overrides },
+    };
+    // The object stays where it is and how it looks: the new local position and, when the parent
+    // chain's turn or size changed, transform rows that absorb the difference.
+    const world = worldTransform(project, holder, object);
+    const before = parentTransform(project, holder, object.parentId).m;
     object.parentId = this.parentId;
-    object.position = localPositionFor(holder, this.parentId, world);
+    object.position = localPositionFor(project, holder, this.parentId, positionOf(world));
+    carryTransform(
+      requireAsset(project, object.assetId),
+      object,
+      before,
+      parentTransform(project, holder, this.parentId).m,
+    );
   }
 
   undo(project: Project): void {
     const object = requireObject(project, this.holderId, this.objectId);
     object.parentId = this.previous.parentId;
     object.position = { ...this.previous.position };
+    object.overrides = { ...this.previous.overrides };
   }
 }
diff --git a/src/core/graph/NodeContext.ts b/src/core/graph/NodeContext.ts
index af47906e..a1f88519 100644
--- a/src/core/graph/NodeContext.ts
+++ b/src/core/graph/NodeContext.ts
@@ -1,6 +1,8 @@
-import type { PropertyBag } from '../model/Properties';
-import { sumAlongChain } from '../model/walk';
+import type { Mat } from '../model/matrix';
+import type { Position, PropertyBag } from '../model/Properties';
+import { type ChainTransform, composeAlongChain } from '../model/walk';
 import { enabledAlongChain } from '../properties/enabled';
+import { localMatrixOf, positionOf, transformOf } from '../properties/transform';
 
 /**
  * Marks a live entity apart from a struct value: both reach a data edge as an object. A symbol
@@ -26,14 +28,23 @@ export interface EntityHandle {
   parent: EntityHandle | null;
 }
 
-export function worldPositionOf(handle: EntityHandle): { x: number; y: number; z: number } {
-  return sumAlongChain(
+/** Position, rotation, scale and mirror of every ancestor applied, nearest last. */
+export function worldTransformOf(handle: EntityHandle): ChainTransform {
+  return composeAlongChain(
     handle,
     (h) => h.parent,
-    (h) => h,
+    (h) => ({ m: localMatrixOf(h, transformOf(h.props)), z: h.z }),
   );
 }
 
+export function worldMatrixOf(handle: EntityHandle): Mat {
+  return worldTransformOf(handle).m;
+}
+
+export function worldPositionOf(handle: EntityHandle): Position {
+  return positionOf(worldTransformOf(handle));
+}
+
 /** Whether a handle and every ancestor of it are switched on. */
 export function isHandleEnabled(handle: EntityHandle): boolean {
   return enabledAlongChain(
diff --git a/src/core/model/hierarchy.ts b/src/core/model/hierarchy.ts
index 44c0c97b..205addd4 100644
--- a/src/core/model/hierarchy.ts
+++ b/src/core/model/hierarchy.ts
@@ -1,7 +1,6 @@
 import { type Removed, removeWhere } from '../util/list';
-import type { Position } from './Properties';
 import type { SceneObject } from './SceneObject';
-import { descendantsBelow, parentChain, sumAlongChain } from './walk';
+import { descendantsBelow, parentChain } from './walk';
 
 /**
  * `objects` stays a flat list in creation order; the tree is implied by `SceneObject.parentId`.
@@ -78,37 +77,6 @@ export function subtreeRoots(tree: ObjectTree, ids: readonly string[]): string[]
   return ids.filter((id) => !ancestorIdsOf(tree, id).some((ancestorId) => set.has(ancestorId)));
 }
 
-function positionAlongChain(tree: ObjectTree, object: SceneObject, withEditorOffset: boolean): Position {
-  return sumAlongChain(
-    object,
-    (o) => parentOf(tree, o),
-    (o) =>
-      withEditorOffset
-        ? { x: o.position.x + o.editorOffset.x, y: o.position.y + o.editorOffset.y, z: o.position.z }
-        : o.position,
-  );
-}
-
-export function worldPosition(tree: ObjectTree, object: SceneObject): Position {
-  return positionAlongChain(tree, object, false);
-}
-
-/**
- * World position plus the editor offsets along the chain. A view aid only – the game, collisions
- * and the graph all use `worldPosition`.
- */
-export function editorPosition(tree: ObjectTree, object: SceneObject): Position {
-  return positionAlongChain(tree, object, true);
-}
-
-/** The local position that puts an object at `world` under the given parent ('' = top level). */
-export function localPositionFor(tree: ObjectTree, parentId: string, world: Position): Position {
-  const parent = parentId ? findObject(tree, parentId) : undefined;
-  if (!parent) return { ...world };
-  const origin = worldPosition(tree, parent);
-  return { x: world.x - origin.x, y: world.y - origin.y, z: world.z - origin.z };
-}
-
 export interface TreeEntry {
   object: SceneObject;
   depth: number;
diff --git a/src/core/model/matrix.ts b/src/core/model/matrix.ts
new file mode 100644
index 00000000..b6140a39
--- /dev/null
+++ b/src/core/model/matrix.ts
@@ -0,0 +1,143 @@
+import type { Vec2 } from './Properties';
+
+/**
+ * A 2D affine transform in Pixi's layout: x' = a·x + c·y + tx, y' = b·x + d·y + ty. Every transform
+ * in the hierarchy is a similarity (uniform scale, rotation, an optional mirror), so their products
+ * are too – which is what lets `decompose` recover rotation and scale from one.
+ */
+export interface Mat {
+  a: number;
+  b: number;
+  c: number;
+  d: number;
+  tx: number;
+  ty: number;
+}
+
+export const IDENTITY: Mat = { a: 1, b: 0, c: 0, d: 1, tx: 0, ty: 0 };
+
+export function translation(x: number, y: number): Mat {
+  return { a: 1, b: 0, c: 0, d: 1, tx: x, ty: y };
+}
+
+/**
+ * Translate ∘ rotate ∘ scale: a point is scaled (a mirror flips the axis first), then turned, then
+ * moved. Rotation in degrees, positive clockwise on screen because y grows downwards.
+ */
+export function transformMatrix(
+  x: number,
+  y: number,
+  rotation: number,
+  scale: number,
+  mirrorX: boolean,
+  mirrorY: boolean,
+): Mat {
+  const r = (rotation * Math.PI) / 180;
+  const cos = Math.cos(r);
+  const sin = Math.sin(r);
+  const sx = mirrorX ? -scale : scale;
+  const sy = mirrorY ? -scale : scale;
+  return { a: cos * sx, b: sin * sx, c: -sin * sy, d: cos * sy, tx: x, ty: y };
+}
+
+/** `outer ∘ inner`: apply `inner` first. */
+export function multiply(outer: Mat, inner: Mat): Mat {
+  return {
+    a: outer.a * inner.a + outer.c * inner.b,
+    b: outer.b * inner.a + outer.d * inner.b,
+    c: outer.a * inner.c + outer.c * inner.d,
+    d: outer.b * inner.c + outer.d * inner.d,
+    tx: outer.a * inner.tx + outer.c * inner.ty + outer.tx,
+    ty: outer.b * inner.tx + outer.d * inner.ty + outer.ty,
+  };
+}
+
+export function apply(m: Mat, p: Vec2): Vec2 {
+  return { x: m.a * p.x + m.c * p.y + m.tx, y: m.b * p.x + m.d * p.y + m.ty };
+}
+
+/** The linear part only – for directions and deltas, which no translation touches. */
+export function applyVector(m: Mat, v: Vec2): Vec2 {
+  return { x: m.a * v.x + m.c * v.y, y: m.b * v.x + m.d * v.y };
+}
+
+export function linearOf(m: Mat): Mat {
+  return { ...m, tx: 0, ty: 0 };
+}
+
+export function determinant(m: Mat): number {
+  return m.a * m.d - m.b * m.c;
+}
+
+/** A transform with scale 0 has no inverse; its translation is undone and the rest left alone. */
+export function invert(m: Mat): Mat {
+  const det = determinant(m);
+  if (Math.abs(det) < 1e-12) return translation(-m.tx, -m.ty);
+  const a = m.d / det;
+  const b = -m.b / det;
+  const c = -m.c / det;
+  const d = m.a / det;
+  return { a, b, c, d, tx: -(a * m.tx + c * m.ty), ty: -(b * m.tx + d * m.ty) };
+}
+
+/** The uniform scale factor of a similarity. */
+export function scaleOf(m: Mat): number {
+  return Math.hypot(m.a, m.b);
+}
+
+/** The angle the x axis was turned by, in radians. */
+export function angleOf(m: Mat): number {
+  return Math.atan2(m.b, m.a);
+}
+
+export function isMirrored(m: Mat): boolean {
+  return determinant(m) < 0;
+}
+
+/**
+ * Rotation (degrees), scale and mirror that rebuild the linear part of a similarity. A mirror
+ * always comes out as `mirrorX`: mirroring y is mirroring x turned by a half circle.
+ */
+export function decompose(m: Mat): { rotation: number; scale: number; mirrorX: boolean } {
+  const mirrorX = isMirrored(m);
+  const a = mirrorX ? -m.a : m.a;
+  const b = mirrorX ? -m.b : m.b;
+  return { rotation: (Math.atan2(b, a) * 180) / Math.PI, scale: scaleOf(m), mirrorX };
+}
+
+export function sameLinear(p: Mat, q: Mat, epsilon = 1e-9): boolean {
+  return (
+    Math.abs(p.a - q.a) < epsilon &&
+    Math.abs(p.b - q.b) < epsilon &&
+    Math.abs(p.c - q.c) < epsilon &&
+    Math.abs(p.d - q.d) < epsilon
+  );
+}
+
+/** The four corners of a `w`×`h` box centred at (`cx`, `cy`) in local units, taken through `m`. */
+export function boxCorners(m: Mat, w: number, h: number, cx = 0, cy = 0): Vec2[] {
+  return [
+    apply(m, { x: cx - w / 2, y: cy - h / 2 }),
+    apply(m, { x: cx + w / 2, y: cy - h / 2 }),
+    apply(m, { x: cx + w / 2, y: cy + h / 2 }),
+    apply(m, { x: cx - w / 2, y: cy + h / 2 }),
+  ];
+}
+
+export interface Extent {
+  left: number;
+  top: number;
+  right: number;
+  bottom: number;
+}
+
+export function extentOfPoints(points: readonly Vec2[]): Extent {
+  const extent = { left: Infinity, top: Infinity, right: -Infinity, bottom: -Infinity };
+  for (const p of points) {
+    extent.left = Math.min(extent.left, p.x);
+    extent.top = Math.min(extent.top, p.y);
+    extent.right = Math.max(extent.right, p.x);
+    extent.bottom = Math.max(extent.bottom, p.y);
+  }
+  return extent;
+}
diff --git a/src/core/model/walk.ts b/src/core/model/walk.ts
index dcdfde4d..4851df48 100644
--- a/src/core/model/walk.ts
+++ b/src/core/model/walk.ts
@@ -1,4 +1,4 @@
-import type { Position } from './Properties';
+import { IDENTITY, type Mat, multiply } from './matrix';
 
 /**
  * `start` and every ancestor above it, nearest first. A revisited item ends the walk: a cyclic
@@ -23,20 +23,26 @@ export function hasParentCycle<T>(start: T, parentOf: (item: T) => T | null | un
   return Boolean(parentOf(last));
 }
 
-/** Summing local positions gives the world position only because the hierarchy has no rotation or scale. */
-export function sumAlongChain<T>(
+/** A world transform: the matrix places, turns and sizes; `z` is draw order and simply adds up. */
+export interface ChainTransform {
+  m: Mat;
+  z: number;
+}
+
+/** Each item's local transform applied inside its parent's, root outermost. */
+export function composeAlongChain<T>(
   start: T,
   parentOf: (item: T) => T | null | undefined,
-  positionOf: (item: T) => Position,
-): Position {
-  const total: Position = { x: 0, y: 0, z: 0 };
+  localOf: (item: T) => ChainTransform,
+): ChainTransform {
+  let m = IDENTITY;
+  let z = 0;
   for (const item of parentChain(start, parentOf)) {
-    const position = positionOf(item);
-    total.x += position.x;
-    total.y += position.y;
-    total.z += position.z;
+    const local = localOf(item);
+    m = multiply(local.m, m);
+    z += local.z;
   }
-  return total;
+  return { m, z };
 }
 
 /** Everything below `rootKey`, depth-first, parents before children. */
diff --git a/src/core/properties/PropertySchema.ts b/src/core/properties/PropertySchema.ts
index f9af0d3c..b2a00d1c 100644
--- a/src/core/properties/PropertySchema.ts
+++ b/src/core/properties/PropertySchema.ts
@@ -96,8 +96,33 @@ export const COMMON_PROPS: PropertySchema = [
   { key: 'enabled', label: 'Enabled', kind: 'boolean', default: true },
 ];
 
-const COMMON_KEYS = new Set(COMMON_PROPS.map((d) => d.key));
+/**
+ * How a drawn or grouping object is placed inside its parent, on top of its position. They cascade
+ * like a position does (see `properties/transform.ts`), so a child turns and grows with its parent.
+ */
+export const TRANSFORM_PROPS: PropertySchema = [
+  { key: 'scale', label: 'Scale', kind: 'number', default: 1, min: 0, step: 0.1 },
+  { key: 'rotation', label: 'Rotation', kind: 'number', default: 0, step: 1 },
+  { key: 'mirrorX', label: 'Mirror X', kind: 'boolean', default: false },
+  { key: 'mirrorY', label: 'Mirror Y', kind: 'boolean', default: false },
+];
 
+/** Cameras and behaviors are placed by the chain but have no look of their own to turn or size. */
+const TRANSFORM_TYPES: ReadonlySet<AssetType> = new Set(['sprite', 'text', 'collider', 'container']);
+
+export function hasTransformProps(type: AssetType): boolean {
+  return TRANSFORM_TYPES.has(type);
+}
+
+const TRANSFORM_KEYS = new Set(TRANSFORM_PROPS.map((d) => d.key));
+
+export function isTransformProp(key: string): boolean {
+  return TRANSFORM_KEYS.has(key);
+}
+
+const COMMON_KEYS = new Set([...COMMON_PROPS, ...TRANSFORM_PROPS].map((d) => d.key));
+
+/** The switch and the transform rows: reserved names, shown apart from the type's own properties. */
 export function isCommonProp(key: string): boolean {
   return COMMON_KEYS.has(key);
 }
@@ -203,7 +228,9 @@ const ASSET_SCHEMAS: Record<AssetType, PropertySchema> = {
 const FULL_SCHEMAS: Record<AssetType, PropertySchema> = Object.fromEntries(
   (Object.keys(ASSET_SCHEMAS) as AssetType[]).map((type) => [
     type,
-    isPlaceableType(type) ? [...COMMON_PROPS, ...ASSET_SCHEMAS[type]] : ASSET_SCHEMAS[type],
+    isPlaceableType(type)
+      ? [...COMMON_PROPS, ...(hasTransformProps(type) ? TRANSFORM_PROPS : []), ...ASSET_SCHEMAS[type]]
+      : ASSET_SCHEMAS[type],
   ]),
 ) as Record<AssetType, PropertySchema>;
 
diff --git a/src/core/properties/bounds.ts b/src/core/properties/bounds.ts
index b42ec73a..9c3933fc 100644
--- a/src/core/properties/bounds.ts
+++ b/src/core/properties/bounds.ts
@@ -1,8 +1,19 @@
 /**
  * Footprints in world units, shared by the viewport, by pointer hit tests in the running game and by
- * "Add collider", so all three agree on how big a thing is.
+ * "Add collider", so all three agree on how big a thing is – and, through the object's transform, on
+ * where its corners are.
  */
-import type { AssetType } from '../model/AssetTypes';
+import { type AssetType, isSymbolType } from '../model/AssetTypes';
+import {
+  apply,
+  boxCorners,
+  type Extent,
+  extentOfPoints,
+  invert,
+  type Mat,
+  scaleOf,
+  translation,
+} from '../model/matrix';
 import { type PropertyBag, textOf, type Vec2 } from '../model/Properties';
 import { cameraFrame } from './camera';
 import { colliderSize } from './collider';
@@ -53,9 +64,34 @@ export function footprintOf(type: AssetType, props: PropertyBag): { w: number; h
   }
 }
 
-export function footprintContains(type: AssetType, props: PropertyBag, centre: Vec2, point: Vec2): boolean {
+/**
+ * How an item of this type is drawn: with its whole transform, or – for the icons of behaviors,
+ * containers and the camera frame – upright and unscaled at its position, since an icon that turns
+ * with its parent would only be harder to read. Colliders are symbols too but follow their body.
+ */
+export function placementOf(type: AssetType, m: Mat): Mat {
+  return isSymbolType(type) && type !== 'collider' ? translation(m.tx, m.ty) : m;
+}
+
+function anchorOffset(type: AssetType, props: PropertyBag, w: number): number {
+  return type === 'text' ? (0.5 - textAnchor(props)) * w : 0;
+}
+
+/** The footprint's corners in world units, grown by `pad` world units on every side. */
+export function footprintCorners(type: AssetType, props: PropertyBag, m: Mat, pad = 0): Vec2[] {
+  const placed = placementOf(type, m);
   const { w, h } = footprintOf(type, props);
-  const offset = type === 'text' ? (0.5 - textAnchor(props)) * w : 0;
-  const x = centre.x + offset;
-  return Math.abs(point.x - x) <= w / 2 && Math.abs(point.y - centre.y) <= h / 2;
+  const local = pad / (scaleOf(placed) || 1);
+  return boxCorners(placed, w + 2 * local, h + 2 * local, anchorOffset(type, props, w), 0);
+}
+
+/** The world rectangle around the drawn footprint – what culling, framing and labels need. */
+export function footprintExtent(type: AssetType, props: PropertyBag, m: Mat): Extent {
+  return extentOfPoints(footprintCorners(type, props, m));
+}
+
+export function footprintContains(type: AssetType, props: PropertyBag, m: Mat, point: Vec2): boolean {
+  const { w, h } = footprintOf(type, props);
+  const local = apply(invert(placementOf(type, m)), point);
+  return Math.abs(local.x - anchorOffset(type, props, w)) <= w / 2 && Math.abs(local.y) <= h / 2;
 }
diff --git a/src/core/properties/collider.ts b/src/core/properties/collider.ts
index 5a2804ed..d01a1083 100644
--- a/src/core/properties/collider.ts
+++ b/src/core/properties/collider.ts
@@ -1,14 +1,18 @@
 /**
  * Collider geometry in world units, shared by the editor and the runtime so both agree on what
- * "overlaps" means.
+ * "overlaps" means. A box is oriented: it turns and grows with the body it sits on.
  */
+import { angleOf, type Mat, scaleOf } from '../model/matrix';
 import type { PropertyBag, Vec2 } from '../model/Properties';
 import { defaultsFor } from './PropertySchema';
 
 const DEFAULTS = defaultsFor('collider');
 
+/** Overlaps this shallow count as touching, so two rotated boxes at rest do not jitter. */
+const EPSILON = 1e-9;
+
 export type ColliderShape =
-  | { kind: 'box'; x: number; y: number; w: number; h: number }
+  | { kind: 'box'; x: number; y: number; w: number; h: number; angle: number }
   | { kind: 'circle'; x: number; y: number; r: number };
 
 type Box = Extract<ColliderShape, { kind: 'box' }>;
@@ -17,20 +21,23 @@ type Circle = Extract<ColliderShape, { kind: 'circle' }>;
 /** A movement axis in the plane (`z` is draw order, not geometry). */
 export type PlaneAxis = 'x' | 'y';
 
-export function colliderShape(props: PropertyBag, centre: Vec2): ColliderShape {
+/** The shape as it sits in the world: sized by the transform's scale, a box turned by its angle. */
+export function colliderShape(props: PropertyBag, m: Mat): ColliderShape {
+  const scale = scaleOf(m);
   if (props.shape === 'circle')
     return {
       kind: 'circle',
-      x: centre.x,
-      y: centre.y,
-      r: Math.max(0, Number(props.radius ?? DEFAULTS.radius)),
+      x: m.tx,
+      y: m.ty,
+      r: Math.max(0, Number(props.radius ?? DEFAULTS.radius)) * scale,
     };
   return {
     kind: 'box',
-    x: centre.x,
-    y: centre.y,
-    w: Math.max(0, Number(props.width ?? DEFAULTS.width)),
-    h: Math.max(0, Number(props.height ?? DEFAULTS.height)),
+    x: m.tx,
+    y: m.ty,
+    w: Math.max(0, Number(props.width ?? DEFAULTS.width)) * scale,
+    h: Math.max(0, Number(props.height ?? DEFAULTS.height)) * scale,
+    angle: angleOf(m),
   };
 }
 
@@ -39,75 +46,143 @@ export function isTrigger(props: PropertyBag): boolean {
   return Boolean(props.isTrigger ?? DEFAULTS.isTrigger);
 }
 
+/** The size in the collider's own units, before any transform. */
 export function colliderSize(props: PropertyBag): { w: number; h: number } {
-  const shape = colliderShape(props, { x: 0, y: 0 });
-  return shape.kind === 'circle' ? { w: shape.r * 2, h: shape.r * 2 } : { w: shape.w, h: shape.h };
+  if (props.shape === 'circle') {
+    const d = Math.max(0, Number(props.radius ?? DEFAULTS.radius)) * 2;
+    return { w: d, h: d };
+  }
+  return {
+    w: Math.max(0, Number(props.width ?? DEFAULTS.width)),
+    h: Math.max(0, Number(props.height ?? DEFAULTS.height)),
+  };
+}
+
+function axesOf(box: Box): [Vec2, Vec2] {
+  const cos = Math.cos(box.angle);
+  const sin = Math.sin(box.angle);
+  return [
+    { x: cos, y: sin },
+    { x: -sin, y: cos },
+  ];
+}
+
+function dot(a: Vec2, b: Vec2): number {
+  return a.x * b.x + a.y * b.y;
+}
+
+/** A world point in the box's own frame: origin at its centre, axes along its sides. */
+function toBoxFrame(box: Box, p: Vec2): Vec2 {
+  const [ux, uy] = axesOf(box);
+  const d = { x: p.x - box.x, y: p.y - box.y };
+  return { x: dot(d, ux), y: dot(d, uy) };
+}
+
+/** Half of the box's projection onto the unit direction `n`. */
+function reach(box: Box, n: Vec2): number {
+  const [ux, uy] = axesOf(box);
+  return (box.w / 2) * Math.abs(dot(ux, n)) + (box.h / 2) * Math.abs(dot(uy, n));
+}
+
+/** How wide the shape is along the unit direction `n` – a turned box measures its own slant. */
+export function shapeExtent(shape: ColliderShape, n: Vec2): number {
+  return shape.kind === 'circle' ? shape.r * 2 : reach(shape, n) * 2;
 }
 
 export function shapeContains(shape: ColliderShape, point: Vec2): boolean {
   if (shape.kind === 'circle')
     return (point.x - shape.x) ** 2 + (point.y - shape.y) ** 2 <= shape.r * shape.r;
-  return Math.abs(point.x - shape.x) * 2 <= shape.w && Math.abs(point.y - shape.y) * 2 <= shape.h;
+  const local = toBoxFrame(shape, point);
+  return Math.abs(local.x) * 2 <= shape.w && Math.abs(local.y) * 2 <= shape.h;
+}
+
+/** Distance from a point to an axis-aligned box centred at the origin; 0 inside. */
+function distanceToBox(p: Vec2, hw: number, hh: number): number {
+  const dx = Math.max(Math.abs(p.x) - hw, 0);
+  const dy = Math.max(Math.abs(p.y) - hh, 0);
+  return Math.hypot(dx, dy);
 }
 
 /** Shapes that merely touch along an edge do not count as overlapping. */
 export function shapesOverlap(a: ColliderShape, b: ColliderShape): boolean {
   if (a.kind === 'box' && b.kind === 'box') {
-    return Math.abs(a.x - b.x) * 2 < a.w + b.w && Math.abs(a.y - b.y) * 2 < a.h + b.h;
+    // Separating axes: the sides of both boxes. Apart along any one of them is apart.
+    const centres = { x: a.x - b.x, y: a.y - b.y };
+    for (const n of [...axesOf(a), ...axesOf(b)])
+      if (reach(a, n) + reach(b, n) - Math.abs(dot(centres, n)) <= EPSILON) return false;
+    return true;
   }
   if (a.kind === 'circle' && b.kind === 'circle') {
     const r = a.r + b.r;
     return (a.x - b.x) ** 2 + (a.y - b.y) ** 2 < r * r;
   }
   const { box, circle } = boxAndCircle(a, b);
-  // Closest point of the box to the circle centre, then a plain distance check.
-  const cx = Math.max(box.x - box.w / 2, Math.min(circle.x, box.x + box.w / 2));
-  const cy = Math.max(box.y - box.h / 2, Math.min(circle.y, box.y + box.h / 2));
-  return (circle.x - cx) ** 2 + (circle.y - cy) ** 2 < circle.r * circle.r;
+  return distanceToBox(toBoxFrame(box, circle), box.w / 2, box.h / 2) < circle.r - EPSILON;
 }
 
 /**
- * The centre distance along `axis` at which the shapes exactly touch, given their current offset on
- * the other axis; 0 when they do not overlap on that other axis and so cannot collide along `axis`.
+ * How far `mover` has to travel along `direction` until it just clears `obstacle`; 0 when they are
+ * apart already. This is what stops a moving object at a wall: it is pushed back against its
+ * travel by exactly this much.
  */
-export function axisSeparation(a: ColliderShape, b: ColliderShape, axis: PlaneAxis): number {
-  const other: PlaneAxis = axis === 'x' ? 'y' : 'x';
-  const offset = Math.abs(a[other] - b[other]);
-  if (a.kind === 'box' && b.kind === 'box') {
-    return offset * 2 < extent(a, other) + extent(b, other) ? (extent(a, axis) + extent(b, axis)) / 2 : 0;
+export function exitDistance(mover: ColliderShape, obstacle: ColliderShape, direction: Vec2): number {
+  const length = Math.hypot(direction.x, direction.y);
+  if (length === 0 || !shapesOverlap(mover, obstacle)) return 0;
+  const d = { x: direction.x / length, y: direction.y / length };
+  if (mover.kind === 'box' && obstacle.kind === 'box') return boxesExit(mover, obstacle, d);
+  if (mover.kind === 'circle' && obstacle.kind === 'circle') {
+    const sep = { x: mover.x - obstacle.x, y: mover.y - obstacle.y };
+    const r = mover.r + obstacle.r;
+    const along = dot(sep, d);
+    return -along + Math.sqrt(Math.max(0, along * along - dot(sep, sep) + r * r));
   }
-  if (a.kind === 'circle' && b.kind === 'circle') {
-    const r = a.r + b.r;
-    return offset < r ? Math.sqrt(r * r - offset * offset) : 0;
-  }
-  const { box, circle } = boxAndCircle(a, b);
-  const halfOther = extent(box, other) / 2;
-  const halfAxis = extent(box, axis) / 2;
-  // Circle centre within the box's span on the other axis: the flat side is hit.
-  if (offset <= halfOther) return halfAxis + circle.r;
-  // Otherwise the corner is the closest point: a circle-vs-point solution.
-  const corner = offset - halfOther;
-  return corner < circle.r ? halfAxis + Math.sqrt(circle.r * circle.r - corner * corner) : 0;
+  const { box, circle } = boxAndCircle(mover, obstacle);
+  // Seen from the box: the circle's centre travels along `d` (or against it when the box moves).
+  const [ux, uy] = axesOf(box);
+  const sign = mover.kind === 'circle' ? 1 : -1;
+  const v = { x: sign * dot(d, ux), y: sign * dot(d, uy) };
+  const start = toBoxFrame(box, circle);
+  return circleExit(start, v, box.w / 2, box.h / 2, circle.r);
 }
 
 /**
- * How far `mover` has to be pushed back against its travel direction `dir` (±1) so it just touches
- * `obstacle`. Only valid when `mover` did not already overlap before its move.
+ * The travel along `d` at which the first separating axis opens – for two boxes that is the exact
+ * moment they stop overlapping. Along an axis the boxes may be moving apart or, when the mover is
+ * already past the obstacle's centre, deeper in and out the far side.
  */
-export function axisCorrection(
-  mover: ColliderShape,
-  obstacle: ColliderShape,
-  axis: PlaneAxis,
-  dir: 1 | -1,
-): number {
-  if (!shapesOverlap(mover, obstacle)) return 0;
-  return Math.max(0, dir * (mover[axis] - obstacle[axis]) + axisSeparation(mover, obstacle, axis));
+function boxesExit(mover: Box, obstacle: Box, d: Vec2): number {
+  const centres = { x: mover.x - obstacle.x, y: mover.y - obstacle.y };
+  let best = Infinity;
+  for (const n of [...axesOf(mover), ...axesOf(obstacle)]) {
+    const speed = dot(d, n);
+    if (Math.abs(speed) < 1e-12) continue;
+    const span = reach(mover, n) + reach(obstacle, n);
+    const offset = dot(centres, n);
+    const apart = offset === 0 || Math.sign(speed) === Math.sign(offset);
+    const t = (apart ? span - Math.abs(offset) : span + Math.abs(offset)) / Math.abs(speed);
+    if (t < best) best = t;
+  }
+  return Number.isFinite(best) ? best : 0;
+}
+
+/**
+ * A circle's centre starting at `start` (inside the box grown by `r`) moving along `v` in the box's
+ * frame: the travel at which it is `r` away from the box. The distance to a box is convex along a
+ * line, so the crossing is found by bisection – exact for sides and corners alike.
+ */
+function circleExit(start: Vec2, v: Vec2, hw: number, hh: number, r: number): number {
+  const outside = (t: number) => distanceToBox({ x: start.x + v.x * t, y: start.y + v.y * t }, hw, hh) - r;
+  let hi = 1;
+  for (let i = 0; i < 64 && outside(hi) <= 0; i++) hi *= 2;
+  let lo = 0;
+  for (let i = 0; i < 60; i++) {
+    const mid = (lo + hi) / 2;
+    if (outside(mid) <= 0) lo = mid;
+    else hi = mid;
+  }
+  return hi;
 }
 
 function boxAndCircle(a: ColliderShape, b: ColliderShape): { box: Box; circle: Circle } {
   return a.kind === 'box' ? { box: a, circle: b as Circle } : { box: b as Box, circle: a };
 }
-
-function extent(box: Box, axis: PlaneAxis): number {
-  return axis === 'x' ? box.w : box.h;
-}
diff --git a/src/core/properties/transform.ts b/src/core/properties/transform.ts
new file mode 100644
index 00000000..5146d959
--- /dev/null
+++ b/src/core/properties/transform.ts
@@ -0,0 +1,151 @@
+/**
+ * Where an object is in the world: its position, rotation, scale and mirror applied inside its
+ * parent's, all the way up (`composeAlongChain`). The model side reads the transform rows off the
+ * asset and the object's overrides; the running world reads them off the entity's props.
+ */
+import type { Asset } from '../model/Asset';
+import { findObject, type ObjectTree } from '../model/hierarchy';
+import {
+  apply,
+  decompose,
+  determinant,
+  IDENTITY,
+  invert,
+  linearOf,
+  type Mat,
+  multiply,
+  sameLinear,
+  scaleOf,
+  angleOf,
+  transformMatrix,
+} from '../model/matrix';
+import { findAsset, type Project } from '../model/Project';
+import type { Position, PropertyBag, Vec2 } from '../model/Properties';
+import type { SceneObject } from '../model/SceneObject';
+import { type ChainTransform, composeAlongChain } from '../model/walk';
+import { hasTransformProps, TRANSFORM_PROPS } from './PropertySchema';
+
+export interface LocalTransform {
+  rotation: number;
+  scale: number;
+  mirrorX: boolean;
+  mirrorY: boolean;
+}
+
+export const NEUTRAL: LocalTransform = { rotation: 0, scale: 1, mirrorX: false, mirrorY: false };
+
+export function transformOf(props: PropertyBag): LocalTransform {
+  return {
+    rotation: finiteOr(props.rotation, 0),
+    scale: finiteOr(props.scale, 1),
+    mirrorX: props.mirrorX === true,
+    mirrorY: props.mirrorY === true,
+  };
+}
+
+function finiteOr(value: unknown, fallback: number): number {
+  const n = Number(value);
+  return Number.isFinite(n) ? n : fallback;
+}
+
+/** The object's own rows without resolving the whole bag: an override, else the asset's value. */
+export function transformOfObject(asset: Asset, object: SceneObject): LocalTransform {
+  if (!hasTransformProps(asset.type)) return NEUTRAL;
+  const bag: PropertyBag = {};
+  for (const { key } of TRANSFORM_PROPS) {
+    const value = object.overrides[key] ?? asset.props[key];
+    if (value !== undefined) bag[key] = value;
+  }
+  return transformOf(bag);
+}
+
+export function localMatrixOf(position: Vec2, t: LocalTransform): Mat {
+  return transformMatrix(position.x, position.y, t.rotation, t.scale, t.mirrorX, t.mirrorY);
+}
+
+function localOf(project: Project, object: SceneObject, withEditorOffset: boolean): ChainTransform {
+  const asset = findAsset(project, object.assetId);
+  const t = asset ? transformOfObject(asset, object) : NEUTRAL;
+  const offset = withEditorOffset ? object.editorOffset : { x: 0, y: 0 };
+  const at = { x: object.position.x + offset.x, y: object.position.y + offset.y };
+  return { m: localMatrixOf(at, t), z: object.position.z };
+}
+
+function chain(project: Project, tree: ObjectTree, object: SceneObject, withEditorOffset: boolean) {
+  return composeAlongChain(
+    object,
+    (o) => (o.parentId ? findObject(tree, o.parentId) : undefined),
+    (o) => localOf(project, o, withEditorOffset),
+  );
+}
+
+export function worldTransform(project: Project, tree: ObjectTree, object: SceneObject): ChainTransform {
+  return chain(project, tree, object, false);
+}
+
+/**
+ * The world transform plus the editor offsets along the chain. A view aid only – the game,
+ * collisions and the graph all use `worldTransform`.
+ */
+export function editorTransform(project: Project, tree: ObjectTree, object: SceneObject): ChainTransform {
+  return chain(project, tree, object, true);
+}
+
+export function positionOf({ m, z }: ChainTransform): Position {
+  return { x: m.tx, y: m.ty, z };
+}
+
+export function worldPosition(project: Project, tree: ObjectTree, object: SceneObject): Position {
+  return positionOf(worldTransform(project, tree, object));
+}
+
+export function editorPosition(project: Project, tree: ObjectTree, object: SceneObject): Position {
+  return positionOf(editorTransform(project, tree, object));
+}
+
+/** The world transform of a parent ('' = top level, the identity). */
+export function parentTransform(project: Project, tree: ObjectTree, parentId: string): ChainTransform {
+  const parent = parentId ? findObject(tree, parentId) : undefined;
+  return parent ? worldTransform(project, tree, parent) : { m: IDENTITY, z: 0 };
+}
+
+/** The local position that puts an object at `world` under the given parent ('' = top level). */
+export function localPositionFor(
+  project: Project,
+  tree: ObjectTree,
+  parentId: string,
+  world: Position,
+): Position {
+  const parent = parentTransform(project, tree, parentId);
+  const local = apply(invert(parent.m), world);
+  return { x: local.x, y: local.y, z: world.z - parent.z };
+}
+
+/**
+ * The rows that keep an object looking the same after the linear part above it changed from
+ * `before` to `after` (re-parenting, grouping). Nothing changes when nothing has to: an object
+ * moved between two upright parents keeps its own rows, mirror flags included.
+ */
+export function carriedTransform(own: LocalTransform, before: Mat, after: Mat): LocalTransform {
+  const carry = multiply(invert(linearOf(after)), linearOf(before));
+  if (sameLinear(carry, IDENTITY)) return own;
+  if (determinant(carry) > 0)
+    return {
+      ...own,
+      rotation: own.rotation + (angleOf(carry) * 180) / Math.PI,
+      scale: own.scale * scaleOf(carry),
+    };
+  return { ...decompose(multiply(carry, localMatrixOf({ x: 0, y: 0 }, own))), mirrorY: false };
+}
+
+/** Writes the carried rows as overrides of a stored object; a type without the rows is left alone. */
+export function carryTransform(asset: Asset, object: SceneObject, before: Mat, after: Mat): void {
+  if (!hasTransformProps(asset.type)) return;
+  const own = transformOfObject(asset, object);
+  const carried = carriedTransform(own, before, after);
+  if (carried === own) return;
+  object.overrides.scale = carried.scale;
+  object.overrides.rotation = carried.rotation;
+  object.overrides.mirrorX = carried.mirrorX;
+  object.overrides.mirrorY = carried.mirrorY;
+}
diff --git a/src/editor/debugHook.ts b/src/editor/debugHook.ts
index 0860a7b2..d14d18dd 100644
--- a/src/editor/debugHook.ts
+++ b/src/editor/debugHook.ts
@@ -187,6 +187,8 @@ export function installDebugHook(app: Application, state: EditorState, editor: E
     shownLabel,
     tabBounds: () => editor.tabBounds(),
     viewportCenter: () => centreOf(viewport),
+    /** Pans the Scene tab to a world point – a step's own lane, away from what earlier steps left. */
+    lookAt: (x: number, y: number) => viewport.lookAt(x, y),
     /** Its drawn box – a step that wants an empty spot must stay inside what the console left over. */
     viewportRect: () => {
       const p = viewport.getGlobalPosition();
diff --git a/src/editor/inspector/objectSection.ts b/src/editor/inspector/objectSection.ts
index 2d31327c..4e26e3cb 100644
--- a/src/editor/inspector/objectSection.ts
+++ b/src/editor/inspector/objectSection.ts
@@ -28,6 +28,7 @@ import type { SceneObject } from '../../core/model/SceneObject';
 import {
   COMMON_PROPS,
   isCommonProp,
+  isTransformProp,
   type PropertyDef,
   schemaFor,
   schemaForObject,
@@ -155,6 +156,11 @@ export function buildObjectSection(host: InspectorHost, holderId: string, object
     });
   }
 
+  // Turn, size and mirror sit with the position: together they are where the object is.
+  const transform = schemaForObject(project, asset, object).filter((def) => isTransformProp(def.key));
+  if (transform.length > 0) addTitle(host, 'TRANSFORM');
+  for (const def of transform) propertyRow(host, holderId, objectId, asset, def, resolved);
+
   // Where the object is *drawn* in the viewport, nothing else – for pulling stacked objects apart.
   addTitle(host, 'EDITOR OFFSET (SCENE VIEW ONLY)');
   for (const axis of OFFSET_AXES) {
diff --git a/src/editor/viewport/SceneViewport.ts b/src/editor/viewport/SceneViewport.ts
index c5acd75c..a43d7cf9 100644
--- a/src/editor/viewport/SceneViewport.ts
+++ b/src/editor/viewport/SceneViewport.ts
@@ -9,10 +9,12 @@ import { MoveSceneObjectCommand, PlaceAssetCommand } from '../../core/commands/s
 import { isSymbolType } from '../../core/model/AssetTypes';
 import { findObject, subtreeRoots } from '../../core/model/hierarchy';
 import { objectRef, resolvedHolder } from '../../core/model/instance';
+import { applyVector, type Extent, invert, linearOf, type Mat } from '../../core/model/matrix';
 import { placementRejection } from '../../core/model/placement';
 import type { Position, Vec2 } from '../../core/model/Properties';
 import { findAsset, type ObjectHolder } from '../../core/model/Project';
-import { footprintOf } from '../../core/properties/bounds';
+import { footprintCorners, footprintExtent } from '../../core/properties/bounds';
+import { localMatrixOf, parentTransform, transformOf } from '../../core/properties/transform';
 import { boundsOf } from '../framing';
 import { type EditorState, objectSelection } from '../state';
 import {
@@ -22,7 +24,6 @@ import {
   drawGrid,
   endDrag,
   onDragChange,
-  type Size,
   Theme,
   textStyle,
   zoomFactor,
@@ -45,7 +46,8 @@ const FIT_MARGIN = 40;
 
 interface ObjectDrag {
   holderId: string;
-  items: { objectId: string; start: Position }[];
+  /** `toLocal` turns a world delta into one in the object's parent frame (the parent may be turned or sized). */
+  items: { objectId: string; start: Position; toLocal: Mat }[];
   pointer: Vec2;
   moved: boolean;
 }
@@ -135,8 +137,8 @@ export class SceneViewport extends WorldViewport {
   fit(): void {
     const { items } = this.collectFrame();
     const boxes = items.map((item) => {
-      const { w, h } = footprintOf(item.type, item.props);
-      return { id: item.id, x: item.x - w / 2, y: item.y - h / 2, w, h };
+      const e = footprintExtent(item.type, item.props, item.m);
+      return { id: item.id, x: e.left, y: e.top, w: e.right - e.left, h: e.bottom - e.top };
     });
     const { minX, minY, maxX, maxY } = boundsOf(boxes, { minX: 0, minY: 0, maxX: 0, maxY: 0 })!;
     const width = Math.max(maxX - minX, 1);
@@ -153,6 +155,13 @@ export class SceneViewport extends WorldViewport {
     this.redraw();
   }
 
+  /** Centres the view on a world point, keeping the zoom (test hook). */
+  lookAt(x: number, y: number): void {
+    this.view.x = x;
+    this.view.y = y;
+    this.redraw();
+  }
+
   protected override currentView(): View {
     return freeView(this.view.x, this.view.y, this.view.zoom, this.w, this.h);
   }
@@ -210,11 +219,11 @@ export class SceneViewport extends WorldViewport {
       this.dragging.moved = true;
       const holder = this.holder();
       if (!holder) return;
-      // A world-space delta is a local-space delta as well (no rotation/scale in the hierarchy).
       for (const item of this.dragging.items) {
+        const local = applyVector(item.toLocal, { x: dx, y: dy });
         objectRef(this.state.project, holder, item.objectId)?.setPosition({
-          x: Math.round(item.start.x + dx),
-          y: Math.round(item.start.y + dy),
+          x: Math.round(item.start.x + local.x),
+          y: Math.round(item.start.y + local.y),
           z: item.start.z,
         });
       }
@@ -331,7 +340,11 @@ export class SceneViewport extends WorldViewport {
     const items = subtreeRoots(holder, this.state.selectedObjectIds)
       .map((id) => findObject(holder, id))
       .filter((o): o is NonNullable<typeof o> => Boolean(o))
-      .map((o) => ({ objectId: o.id, start: { ...o.position } }));
+      .map((o) => ({
+        objectId: o.id,
+        start: { ...o.position },
+        toLocal: invert(linearOf(parentTransform(this.state.project, holder, o.parentId).m)),
+      }));
     const local = this.toLocal(e.global);
     this.dragging = { holderId: holder.id, items, pointer: { x: local.x, y: local.y }, moved: false };
   }
@@ -368,8 +381,8 @@ export class SceneViewport extends WorldViewport {
     this.drawGizmos(items, view);
   }
 
-  protected override syncDecorations(item: DrawItem, onScreen: boolean, scale: number, size: Size): void {
-    this.syncLabel(item, onScreen, scale, size);
+  protected override syncDecorations(item: DrawItem, onScreen: boolean, scale: number, extent: Extent): void {
+    this.syncLabel(item, onScreen, scale, extent);
   }
 
   protected override disposeDecorations(id: string): void {
@@ -411,7 +424,7 @@ export class SceneViewport extends WorldViewport {
     }
   }
 
-  private syncLabel(item: DrawItem, show: boolean, scale: number, size: Size): void {
+  private syncLabel(item: DrawItem, show: boolean, scale: number, extent: Extent): void {
     let label = this.labels.get(item.id);
     if (!show || !isSymbolType(item.type)) {
       if (label) label.visible = false;
@@ -427,8 +440,8 @@ export class SceneViewport extends WorldViewport {
     label.text = item.name;
     label.scale.set(1 / scale);
     if (item.type === 'camera' || item.type === 'collider')
-      label.position.set(item.x - size.w / 2, item.y - size.h / 2 - 16 / scale);
-    else label.position.set(item.x + size.w / 2 + 4 / scale, item.y - 8 / scale);
+      label.position.set(extent.left, extent.top - 16 / scale);
+    else label.position.set(extent.right + 4 / scale, item.y - 8 / scale);
   }
 
   /**
@@ -440,12 +453,10 @@ export class SceneViewport extends WorldViewport {
     const asset = preview && findAsset(this.state.project, preview.assetId);
     if (!preview || !asset) return;
     const color = preview.rejection ? Theme.colors.danger : Theme.colors.accent;
-    const { w, h } = footprintOf(asset.type, asset.props);
+    const m = localMatrixOf(preview.at, transformOf(asset.props));
     const centre = worldToScreen(view, preview.at.x, preview.at.y);
-    const sw = Math.max(w * view.scale, 8);
-    const sh = Math.max(h * view.scale, 8);
     this.gizmos
-      .rect(centre.x - sw / 2, centre.y - sh / 2, sw, sh)
+      .poly(this.screenCorners(footprintCorners(asset.type, asset.props, m), view), true)
       .fill({ color, alpha: 0.12 })
       .stroke({ width: 2, color });
     this.gizmos
@@ -456,6 +467,10 @@ export class SceneViewport extends WorldViewport {
       .stroke({ width: 1, color });
   }
 
+  private screenCorners(corners: readonly Vec2[], view: View): Vec2[] {
+    return corners.map((c) => worldToScreen(view, c.x, c.y));
+  }
+
   /** Selection outlines – only the primary object gets corner handles – and the rubber band. */
   private drawGizmos(items: DrawItem[], view: View): void {
     this.lastGizmos = [];
@@ -464,27 +479,20 @@ export class SceneViewport extends WorldViewport {
       for (const id of s.objectIds) {
         const item = items.find((i) => i.id === id);
         if (!item) continue;
-        const { w, h } = footprintOf(item.type, item.props);
-        const tl = worldToScreen(view, item.x - w / 2, item.y - h / 2);
-        const sw = w * view.scale;
-        const sh = h * view.scale;
+        // Two screen pixels outside the footprint, whichever way it is turned.
+        const corners = this.screenCorners(
+          footprintCorners(item.type, item.props, item.m, 2 / view.scale),
+          view,
+        );
         // A switched-off object draws nothing, so its outline is the only way to see where it
         // sits – dimmed, because the empty frame would otherwise read as a drawing that failed.
         const alpha = item.enabled ? 1 : OFF_GIZMO_ALPHA;
         const primary = id === s.objectId;
         this.lastGizmos.push({ id, primary, alpha });
-        this.gizmos
-          .rect(tl.x - 2, tl.y - 2, sw + 4, sh + 4)
-          .stroke({ width: 2, color: Theme.colors.gizmo, alpha });
+        this.gizmos.poly(corners, true).stroke({ width: 2, color: Theme.colors.gizmo, alpha });
         if (!primary) continue;
-        for (const [cx, cy] of [
-          [tl.x, tl.y],
-          [tl.x + sw, tl.y],
-          [tl.x, tl.y + sh],
-          [tl.x + sw, tl.y + sh],
-        ] as const) {
-          this.gizmos.rect(cx - 3, cy - 3, 6, 6).fill({ color: Theme.colors.gizmo, alpha });
-        }
+        for (const c of corners)
+          this.gizmos.rect(c.x - 3, c.y - 3, 6, 6).fill({ color: Theme.colors.gizmo, alpha });
       }
     }
     this.drawDropPreview(view);
diff --git a/src/editor/viewport/WorldViewport.ts b/src/editor/viewport/WorldViewport.ts
index 934c7d88..25fd126a 100644
--- a/src/editor/viewport/WorldViewport.ts
+++ b/src/editor/viewport/WorldViewport.ts
@@ -1,14 +1,21 @@
 import { Container, type FederatedPointerEvent, Graphics, type Mesh, Text, TextStyle } from 'pixi.js';
 import { isSymbolType } from '../../core/model/AssetTypes';
+import { type Extent, scaleOf } from '../../core/model/matrix';
 import { sortByZ, type Vec2, textOf } from '../../core/model/Properties';
-import { footprintOf, textAnchor } from '../../core/properties/bounds';
-import { rectAround, rectsIntersect } from '../../core/properties/camera';
+import { footprintExtent, footprintOf, placementOf, textAnchor } from '../../core/properties/bounds';
+import { rectsIntersect } from '../../core/properties/camera';
 import type { EditorState } from '../state';
 import { type Size, Theme, Widget } from '../../ui';
 import type { DrawItem } from '../../viewport/items';
 import { shaderLibrary } from '../../viewport/ShaderLibrary';
 import { type ShaderSource, wornShader } from '../../viewport/shaders';
-import { disposeSpriteMesh, drawItem, newSpriteMesh, syncSpriteMesh } from '../../viewport/symbols';
+import {
+  disposeSpriteMesh,
+  drawItem,
+  newSpriteMesh,
+  placeNode,
+  syncSpriteMesh,
+} from '../../viewport/symbols';
 import { textureLibrary } from '../../viewport/TextureLibrary';
 import { type View, worldBounds, worldToScreen } from '../../viewport/view';
 
@@ -78,7 +85,8 @@ export abstract class WorldViewport extends Widget {
 
   protected afterItems(_items: DrawItem[], _view: View): void {}
 
-  protected syncDecorations(_item: DrawItem, _onScreen: boolean, _scale: number, _size: Size): void {}
+  /** `extent` is the world rectangle around the item as drawn, turn and size included. */
+  protected syncDecorations(_item: DrawItem, _onScreen: boolean, _scale: number, _extent: Extent): void {}
 
   /** An item disappeared: drop whatever `syncDecorations` created for it. */
   protected disposeDecorations(_id: string): void {}
@@ -120,19 +128,22 @@ export abstract class WorldViewport extends Widget {
     return { x: origin.x + local.x, y: origin.y + local.y };
   }
 
-  /** The screen rectangle an object occupies, so tests can sample the pixels inside it. */
+  /**
+   * The screen rectangle around an object, so tests can sample the pixels inside it: centre plus
+   * size, and for a turned object the box around its corners.
+   */
   objectRect(objectId: string): { x: number; y: number; w: number; h: number } | null {
     const view = this.currentView();
     const item = this.lastItems.find((i) => i.id === objectId);
     if (!item || !view) return null;
-    const local = worldToScreen(view, item.x, item.y);
+    const extent = footprintExtent(item.type, item.props, item.m);
+    const local = worldToScreen(view, (extent.left + extent.right) / 2, (extent.top + extent.bottom) / 2);
     const origin = this.getGlobalPosition();
-    const size = footprintOf(item.type, item.props);
     return {
       x: origin.x + local.x,
       y: origin.y + local.y,
-      w: size.w * view.scale,
-      h: size.h * view.scale,
+      w: (extent.right - extent.left) * view.scale,
+      h: (extent.bottom - extent.top) * view.scale,
     };
   }
 
@@ -164,9 +175,10 @@ export abstract class WorldViewport extends Widget {
     for (const item of items) {
       seen.add(item.id);
       const size = footprintOf(item.type, item.props);
+      const extent = footprintExtent(item.type, item.props, item.m);
       // A switched-off object is not drawn at all, in the scene view as in the game; it is
       // selected and switched back on from the hierarchy panel.
-      const onScreen = item.enabled && rectsIntersect(rectAround(item.x, item.y, size.w, size.h), bounds);
+      const onScreen = item.enabled && rectsIntersect(extent, bounds);
       let node = this.nodes.get(item.id);
       if (!node) {
         node = new Graphics();
@@ -178,13 +190,12 @@ export abstract class WorldViewport extends Widget {
       }
       node.visible = onScreen;
       node.eventMode = this.itemsInteractive && onScreen ? 'static' : 'none';
-      this.syncDecorations(item, onScreen, view.scale, size);
+      this.syncDecorations(item, onScreen, view.scale, extent);
       this.syncText(item, onScreen, view.scale);
       this.syncSprite(item, onScreen, size, shaders);
       if (!onScreen) continue;
       this.visibleCount++;
-      node.position.set(item.x, item.y);
-      drawItem(node, item, size, view.scale);
+      drawItem(node, item, size, view.scale * scaleOf(placementOf(item.type, item.m)));
     }
     this.afterItems(items, view);
     for (const [id, node] of this.nodes) {
@@ -253,7 +264,7 @@ export abstract class WorldViewport extends Widget {
       this.objects.addChild(text);
     }
     text.visible = true;
-    text.position.set(item.x, item.y);
+    placeNode(text, item.m);
     text.text = textOf(item.props.text ?? '');
     const size = Math.max(1, Number(item.props.fontSize ?? 32));
     if (text.style.fontSize !== size) text.style.fontSize = size;
diff --git a/src/runtime/World.ts b/src/runtime/World.ts
index 483d047f..5bb2d64c 100644
--- a/src/runtime/World.ts
+++ b/src/runtime/World.ts
@@ -1,4 +1,4 @@
-import { ENTITY_HANDLE, isHandleEnabled, type PointerState, worldPositionOf } from '../core/graph';
+import { ENTITY_HANDLE, isHandleEnabled, type PointerState, worldMatrixOf } from '../core/graph';
 import { newId } from '../core/ids';
 import type { Asset } from '../core/model/Asset';
 import { isPlaceableType } from '../core/model/AssetTypes';
@@ -253,11 +253,11 @@ export class World {
     const parts = [entity, ...this.descendantsOf(entity)].filter((e) => isHandleEnabled(e));
     const colliders = parts.filter((e) => e.type === 'collider');
     if (colliders.length > 0)
-      return colliders.some((c) => shapeContains(colliderShape(c.props, worldPositionOf(c)), point));
+      return colliders.some((c) => shapeContains(colliderShape(c.props, worldMatrixOf(c)), point));
     return parts.some(
       (e) =>
         (e.type === 'sprite' || e.type === 'text') &&
-        footprintContains(e.type, e.props, worldPositionOf(e), point),
+        footprintContains(e.type, e.props, worldMatrixOf(e), point),
     );
   }
 
diff --git a/src/runtime/systems/CollisionSystem.ts b/src/runtime/systems/CollisionSystem.ts
index c7e824dc..1e6cab48 100644
--- a/src/runtime/systems/CollisionSystem.ts
+++ b/src/runtime/systems/CollisionSystem.ts
@@ -1,11 +1,13 @@
-import { isHandleEnabled, worldPositionOf } from '../../core/graph';
+import { isHandleEnabled, worldMatrixOf } from '../../core/graph';
+import { applyVector, IDENTITY } from '../../core/model/matrix';
+import type { Vec2 } from '../../core/model/Properties';
 import { parentChain } from '../../core/model/walk';
 import {
-  axisCorrection,
   colliderShape,
-  colliderSize,
+  exitDistance,
   isTrigger,
   type PlaneAxis,
+  shapeExtent,
   shapesOverlap,
 } from '../../core/properties/collider';
 import type { Entity } from '../Entity';
@@ -78,6 +80,7 @@ export class CollisionSystem implements System {
   /**
    * Moves one axis at a time, stopping each at contact so the object slides along walls. An existing
    * overlap is never resolved (no jolt) but never deepened, so an object can leave a wall but not burrow in.
+   * The axes are the entity's own: under a turned parent, "x" runs wherever that parent points it.
    */
   moveAndSlide(
     world: World,
@@ -107,12 +110,13 @@ export class CollisionSystem implements System {
     delta: number,
   ): boolean {
     if (delta === 0 || own.length === 0) return false;
+    const travel = axisTravel(entity, axis);
     const steps = Math.min(
       MAX_STEPS,
-      Math.max(1, Math.ceil(Math.abs(delta) / stepLimit(own, obstacles, axis))),
+      Math.max(1, Math.ceil((Math.abs(delta) * travel.length) / stepLimit(own, obstacles, travel.unit))),
     );
     for (let i = 0; i < steps; i++)
-      if (this.stepAxis(entity, own, obstacles, axis, delta / steps)) return true;
+      if (this.stepAxis(entity, own, obstacles, axis, delta / steps, travel)) return true;
     return false;
   }
 
@@ -123,8 +127,10 @@ export class CollisionSystem implements System {
     obstacles: Entity[],
     axis: PlaneAxis,
     delta: number,
+    travel: AxisTravel,
   ): boolean {
     const dir: 1 | -1 = delta > 0 ? 1 : -1;
+    const forward = { x: dir * travel.unit.x, y: dir * travel.unit.y };
     // How deep each pair already sits in the other before the step. A step may never deepen
     // that, but resolving it is not our job (no jolt, and leaving a wall is always possible).
     // Overlaps below the skin count as plain contact – that is where rounding leftovers of a
@@ -132,36 +138,51 @@ export class CollisionSystem implements System {
     const before: [Entity, Entity, number][] = [];
     for (const a of own) {
       for (const b of obstacles) {
-        const p = penetration(a, b, axis, dir);
+        const p = penetration(a, b, forward);
         before.push([a, b, p > SKIN ? p : 0]);
       }
     }
     entity[axis] += delta;
     let correction = 0;
     for (const [a, b, was] of before) {
-      const t = penetration(a, b, axis, dir) - was;
+      const t = penetration(a, b, forward) - was;
       if (t > correction) correction = t;
     }
     if (correction <= 0) return false;
-    entity[axis] -= dir * Math.min(correction, Math.abs(delta));
+    entity[axis] -= dir * Math.min(correction / travel.length, Math.abs(delta));
     return true;
   }
 }
 
+/** Where one unit along an entity's local axis goes in the world, and how far. */
+interface AxisTravel {
+  unit: Vec2;
+  length: number;
+}
+
+function axisTravel(entity: Entity, axis: PlaneAxis): AxisTravel {
+  const parent = entity.parent ? worldMatrixOf(entity.parent) : IDENTITY;
+  const v = applyVector(parent, axis === 'x' ? { x: 1, y: 0 } : { x: 0, y: 1 });
+  const length = Math.hypot(v.x, v.y);
+  return length > 0
+    ? { unit: { x: v.x / length, y: v.y / length }, length }
+    : { unit: { x: 1, y: 0 }, length: 1 };
+}
+
 function shapeOf(collider: Entity) {
-  return colliderShape(collider.props, worldPositionOf(collider));
+  return colliderShape(collider.props, worldMatrixOf(collider));
 }
 
 /**
- * How far `a` would have to be pushed back against `dir` to just clear `b` (0 when they are
- * apart). Once a's centre has passed b's centre along `dir` the shorter way out is forwards,
+ * How far `a` would have to be pushed back against its travel to just clear `b` (0 when they are
+ * apart). Once a's centre has passed b's centre along the travel the shorter way out is forwards,
  * so nothing blocks – that keeps an object that ended up inside something able to leave.
  */
-function penetration(a: Entity, b: Entity, axis: PlaneAxis, dir: 1 | -1): number {
+function penetration(a: Entity, b: Entity, forward: Vec2): number {
   const mover = shapeOf(a);
   const obstacle = shapeOf(b);
-  if (dir * (mover[axis] - obstacle[axis]) >= 0) return 0;
-  return axisCorrection(mover, obstacle, axis, dir);
+  if ((mover.x - obstacle.x) * forward.x + (mover.y - obstacle.y) * forward.y >= 0) return 0;
+  return exitDistance(mover, obstacle, { x: -forward.x, y: -forward.y });
 }
 
 /** Whether `collider` belongs to `entity` or to something parented (at any depth) under it. */
@@ -170,19 +191,17 @@ function travelsWith(collider: Entity, entity: Entity): boolean {
   return false;
 }
 
-/** Longest safe step along `axis`: half of the thinnest collider involved (never below 1). */
-function stepLimit(own: Entity[], obstacles: Entity[], axis: PlaneAxis): number {
+/**
+ * Longest safe step in world units: half of the thinnest collider involved, measured along the
+ * travel as the colliders sit in the world (never below 1).
+ */
+function stepLimit(own: Entity[], obstacles: Entity[], along: Vec2): number {
   let min = Infinity;
-  for (const e of own) min = Math.min(min, extentOf(e, axis));
-  for (const e of obstacles) min = Math.min(min, extentOf(e, axis));
+  for (const e of own) min = Math.min(min, shapeExtent(shapeOf(e), along));
+  for (const e of obstacles) min = Math.min(min, shapeExtent(shapeOf(e), along));
   return Math.max(1, min / 2);
 }
 
-function extentOf(collider: Entity, axis: PlaneAxis): number {
-  const size = colliderSize(collider.props);
-  return axis === 'x' ? size.w : size.h;
-}
-
 /** The object a collider belongs to: its parent, or the collider itself when top-level. */
 export function bodyOf(collider: Entity): Entity {
   return collider.parent ?? collider;
diff --git a/src/viewport/items.ts b/src/viewport/items.ts
index 6d6529de..ec422c98 100644
--- a/src/viewport/items.ts
+++ b/src/viewport/items.ts
@@ -1,12 +1,14 @@
-import { isHandleEnabled, worldPositionOf } from '../core/graph';
+import { isHandleEnabled, worldTransformOf } from '../core/graph';
 import { type AssetType, isSymbolType } from '../core/model/AssetTypes';
-import { editorPosition, treeOrder } from '../core/model/hierarchy';
+import { treeOrder } from '../core/model/hierarchy';
 import { resolvedHolder } from '../core/model/instance';
+import type { Mat } from '../core/model/matrix';
 import type { PropertyBag } from '../core/model/Properties';
 import { findAsset, type ObjectHolder, type Project } from '../core/model/Project';
 import { resolveTexture, type TextureRef } from '../core/model/texture';
 import { enabledAlongChain } from '../core/properties/enabled';
 import { resolveSceneObjectProps } from '../core/properties/resolve';
+import { editorTransform } from '../core/properties/transform';
 import { animationTime } from '../runtime/systems/AnimationSystem';
 import type { World } from '../runtime/World';
 
@@ -19,6 +21,8 @@ export interface DrawItem {
   x: number;
   y: number;
   z: number;
+  /** The whole world transform the position is part of: how the item is turned, sized, mirrored. */
+  m: Mat;
   props: PropertyBag;
   /** Id of the parent item, empty when top-level. */
   parentId: string;
@@ -50,11 +54,15 @@ export function collectHolderItems(
     const asset = findAsset(project, object.assetId);
     if (!asset) continue;
     const props = resolveSceneObjectProps(project, asset, object);
+    const { m, z } = editorTransform(project, view, object);
     items.push({
       id: object.id,
       type: asset.type,
       name: object.name || asset.name,
-      ...editorPosition(view, object),
+      x: m.tx,
+      y: m.ty,
+      z,
+      m,
       props,
       parentId: object.parentId,
       enabled: true,
@@ -75,11 +83,18 @@ export function collectHolderItems(
 export function collectWorldItems(world: World): DrawList {
   const items = world.entities
     .filter((e) => !isSymbolType(e.type) && isHandleEnabled(e))
-    .map((e) => ({
+    .map((e) => {
+      const { m, z } = worldTransformOf(e);
+      return { e, m, z };
+    })
+    .map(({ e, m, z }) => ({
       id: e.objectId,
       type: e.type,
       name: e.name,
-      ...worldPositionOf(e),
+      x: m.tx,
+      y: m.ty,
+      z,
+      m,
       props: e.props,
       parentId: e.parent?.objectId ?? '',
       enabled: true,
diff --git a/src/viewport/symbols.ts b/src/viewport/symbols.ts
index 59d0970e..999702d8 100644
--- a/src/viewport/symbols.ts
+++ b/src/viewport/symbols.ts
@@ -1,6 +1,7 @@
-/** Everything here draws in world units at the item's origin. */
-import { type Graphics, Mesh, MeshGeometry, Rectangle, Texture } from 'pixi.js';
-import { BEHAVIOR_SYMBOL, CONTAINER_SYMBOL, textAnchor } from '../core/properties/bounds';
+/** Everything here draws in the item's own units at its origin; `placeNode` puts it into the world. */
+import { type Container, type Graphics, Mesh, MeshGeometry, Rectangle, Texture } from 'pixi.js';
+import { determinant, IDENTITY, type Mat } from '../core/model/matrix';
+import { BEHAVIOR_SYMBOL, CONTAINER_SYMBOL, placementOf, textAnchor } from '../core/properties/bounds';
 import { type ColliderShape, colliderShape, isTrigger } from '../core/properties/collider';
 import { DEFAULT_EDGE } from '../core/shader/nodes';
 import { Theme } from '../ui';
@@ -11,6 +12,19 @@ import type { TextureLibrary } from './TextureLibrary';
 /** Screen-pixel half-width of the grab band along a collider's outline. */
 const COLLIDER_GRAB_BAND = 6;
 
+/**
+ * Gives a Pixi node the transform `m`, its scale multiplied by `w`×`h` (a unit quad becomes the
+ * sprite's box). A similarity is a rotation and two scales whose signs carry the mirror, which is
+ * exactly what a node has – no skew needed.
+ */
+export function placeNode(node: Container, m: Mat, w = 1, h = 1): void {
+  const sx = Math.hypot(m.a, m.b);
+  const sy = Math.hypot(m.c, m.d) * (determinant(m) < 0 ? -1 : 1);
+  node.position.set(m.tx, m.ty);
+  node.rotation = Math.atan2(m.b, m.a);
+  node.scale.set(sx * w, sy * h);
+}
+
 /**
  * One quad for every sprite, sized by the mesh's scale. A frame is a texture of its own
  * (`TextureLibrary`), so the 0..1 UVs cover exactly the frame.
@@ -44,8 +58,7 @@ export function syncSpriteMesh(
   // the picture is there the colour tints it.
   const edge = wornShader(item.props, shaders)?.edge ?? DEFAULT_EDGE;
   const texture = item.texture ? textures.get(item.texture, edge) : null;
-  mesh.position.set(item.x, item.y);
-  mesh.scale.set(size.w, size.h);
+  placeNode(mesh, item.m, size.w, size.h);
   mesh.texture = texture ?? Texture.WHITE;
   mesh.tint = Number(item.props.color ?? 0xffffff);
   syncShader(mesh, item.props, shaders);
@@ -57,13 +70,17 @@ export function disposeSpriteMesh(mesh: Mesh): void {
   mesh.destroy();
 }
 
-/** `scale` is the view scale in pixels per world unit, so strokes stay constant on screen. */
+/**
+ * Draws in the node's own units after `placeNode` put it where the item is. `scale` is the
+ * on-screen pixels per unit of the node, so strokes stay constant on screen.
+ */
 export function drawItem(
   node: Graphics,
   item: DrawItem,
   size: { w: number; h: number },
   scale: number,
 ): void {
+  placeNode(node, placementOf(item.type, item.m));
   node.clear();
   node.hitArea = null;
   switch (item.type) {
@@ -116,7 +133,7 @@ export function drawItem(
     case 'collider': {
       // Only the outline band is hit-testable, so clicks inside still reach the sprite that the
       // collider sits on.
-      const shape = colliderShape(item.props, { x: 0, y: 0 });
+      const shape = colliderShape(item.props, IDENTITY);
       if (shape.kind === 'circle') node.circle(0, 0, shape.r);
       else node.rect(-shape.w / 2, -shape.h / 2, shape.w, shape.h);
       const color = isTrigger(item.props) ? Theme.colors.trigger : Theme.colors.collider;
diff --git a/tests/core/collider.test.ts b/tests/core/collider.test.ts
index 52aeeeae..7b2c6530 100644
--- a/tests/core/collider.test.ts
+++ b/tests/core/collider.test.ts
@@ -3,16 +3,26 @@ import { SetAssetPropertyCommand } from '../../src/core/commands/assetCommands';
 import { AddColliderCommand } from '../../src/core/commands/colliderCommands';
 import { SetOverrideCommand } from '../../src/core/commands/sceneObjectCommands';
 import { findAsset } from '../../src/core/model/Project';
+import { transformMatrix, translation } from '../../src/core/model/matrix';
 import {
-  axisCorrection,
-  axisSeparation,
   colliderShape,
+  exitDistance,
+  shapeContains,
   shapesOverlap,
 } from '../../src/core/properties/collider';
 import { graphEditor, makeBehavior, makeSpriteScene, place, roundTrip, startWorld } from '../helpers';
 
-const box = (x: number, y: number, w: number, h: number) => ({ kind: 'box' as const, x, y, w, h });
+const box = (x: number, y: number, w: number, h: number, degrees = 0) => ({
+  kind: 'box' as const,
+  x,
+  y,
+  w,
+  h,
+  angle: (degrees * Math.PI) / 180,
+});
 const circle = (x: number, y: number, r: number) => ({ kind: 'circle' as const, x, y, r });
+const LEFT = { x: -1, y: 0 };
+const UP = { x: 0, y: -1 };
 
 describe('collider geometry', () => {
   it('box vs box: overlapping yes, touching edges no', () => {
@@ -21,6 +31,19 @@ describe('collider geometry', () => {
     expect(shapesOverlap(box(0, 0, 100, 100), box(0, 150, 100, 100))).toBe(false);
   });
 
+  it('a turned box reaches where its corners are, not where its upright frame would be', () => {
+    // A 100×20 bar turned by 45° reaches ≈ 42.4 along x; upright it would reach 50.
+    expect(shapesOverlap(box(0, 0, 100, 20, 45), box(95, 0, 100, 100))).toBe(false);
+    expect(shapesOverlap(box(0, 0, 100, 20), box(95, 0, 100, 100))).toBe(true);
+    // A diamond (square turned by 45°, corners at ±70.7) and a square at (110, 110): their upright
+    // frames overlap (120.7 > 110 on both axes) but the square's corner (60, 60) lies outside the
+    // diamond's edge x + y = 70.7.
+    expect(shapesOverlap(box(0, 0, 100, 100, 45), box(110, 110, 100, 100))).toBe(false);
+    expect(shapesOverlap(box(0, 0, 100, 100, 45), box(95, 0, 100, 100))).toBe(true);
+    expect(shapeContains(box(0, 0, 100, 20, 90), { x: 0, y: 45 })).toBe(true);
+    expect(shapeContains(box(0, 0, 100, 20, 90), { x: 45, y: 0 })).toBe(false);
+  });
+
   it('circle vs circle and box vs circle (corner case: the closest point is a box corner)', () => {
     expect(shapesOverlap(circle(0, 0, 10), circle(15, 0, 10))).toBe(true);
     expect(shapesOverlap(circle(0, 0, 10), circle(20, 0, 10))).toBe(false);
@@ -29,28 +52,51 @@ describe('collider geometry', () => {
     // Corner at (50,50): a circle at (60,60) with r=10 has distance √200 ≈ 14.1 → no overlap; r=15 → overlap.
     expect(shapesOverlap(circle(60, 60, 10), box(0, 0, 100, 100))).toBe(false);
     expect(shapesOverlap(circle(60, 60, 15), box(0, 0, 100, 100))).toBe(true);
+    // The same corner turned to (0, 70.7): the circle at (0, 80) with r=10 just misses it.
+    expect(shapesOverlap(circle(0, 80, 10), box(0, 0, 100, 100, 45))).toBe(true);
+    expect(shapesOverlap(circle(0, 81, 10), box(0, 0, 100, 100, 45))).toBe(false);
   });
 
-  it('builds shapes from properties with defaults', () => {
-    expect(colliderShape({}, { x: 1, y: 2 })).toEqual(box(1, 2, 100, 100));
-    expect(colliderShape({ shape: 'circle', radius: 7 }, { x: 0, y: 0 })).toEqual(circle(0, 0, 7));
+  it('builds shapes from properties with defaults, sized and turned by the transform', () => {
+    expect(colliderShape({}, translation(1, 2))).toEqual(box(1, 2, 100, 100));
+    expect(colliderShape({ shape: 'circle', radius: 7 }, translation(0, 0))).toEqual(circle(0, 0, 7));
+    const turned = colliderShape({ width: 40, height: 20 }, transformMatrix(5, 6, 30, 2, false, false));
+    expect(turned).toMatchObject({ kind: 'box', x: 5, y: 6, w: 80, h: 40 });
+    expect(turned.kind === 'box' && (turned.angle * 180) / Math.PI).toBeCloseTo(30);
+    // A mirror keeps a box a box: the same shape, whichever way its x axis points.
+    const mirrored = colliderShape({ width: 40, height: 20 }, transformMatrix(0, 0, 0, 1, true, false));
+    expect(shapeContains(mirrored, { x: 19, y: 9 })).toBe(true);
+    expect(shapeContains(mirrored, { x: 21, y: 9 })).toBe(false);
+    expect(colliderShape({ shape: 'circle', radius: 7 }, transformMatrix(0, 0, 0, 3, false, false))).toEqual(
+      circle(0, 0, 21),
+    );
   });
 
-  it('axis separation: the centre distance at which shapes touch along one axis', () => {
-    expect(axisSeparation(box(0, 0, 100, 60), box(0, 10, 40, 60), 'x')).toBe(70);
-    expect(axisSeparation(box(0, 0, 100, 60), box(0, 100, 40, 60), 'x')).toBe(0); // apart on y
-    expect(axisSeparation(circle(0, 0, 10), circle(0, 12, 10), 'x')).toBeCloseTo(16);
-    expect(axisSeparation(box(0, 0, 100, 100), circle(0, 20, 10), 'x')).toBe(60); // flat side
-    expect(axisSeparation(box(0, 0, 100, 100), circle(0, 56, 10), 'x')).toBeCloseTo(50 + 8); // corner: dy=6 → √(100−36)
-    expect(axisSeparation(box(0, 0, 100, 100), circle(0, 70, 10), 'x')).toBe(0);
-  });
-
-  it('axis correction: how far a mover must back off to just touch', () => {
-    // Mover moved right into a box: centres 80 apart, need 100 → push back 20.
-    expect(axisCorrection(box(0, 0, 100, 100), box(80, 0, 100, 100), 'x', 1)).toBe(20);
-    expect(axisCorrection(box(80, 0, 100, 100), box(0, 0, 100, 100), 'x', -1)).toBe(20);
-    expect(axisCorrection(box(0, 0, 100, 100), box(120, 0, 100, 100), 'x', 1)).toBe(0);
-    expect(axisCorrection(box(0, 0, 100, 100), box(0, 90, 100, 100), 'y', 1)).toBe(10);
+  it('exit distance: how far a mover must travel to just clear an obstacle', () => {
+    // Mover moved right into a box: centres 80 apart, need 100 → 20 back to the left.
+    expect(exitDistance(box(80, 0, 100, 100), box(0, 0, 100, 100), { x: 1, y: 0 })).toBe(20);
+    expect(exitDistance(box(0, 0, 100, 100), box(80, 0, 100, 100), LEFT)).toBe(20);
+    expect(exitDistance(box(0, 0, 100, 100), box(120, 0, 100, 100), LEFT)).toBe(0);
+    expect(exitDistance(box(0, 0, 100, 100), box(0, 90, 100, 100), UP)).toBe(10);
+    expect(exitDistance(box(0, 10, 40, 60), box(0, 0, 100, 60), LEFT)).toBe(70);
+    expect(exitDistance(box(0, 100, 40, 60), box(0, 0, 100, 60), LEFT)).toBe(0); // apart on y
+    expect(exitDistance(circle(0, 12, 10), circle(0, 0, 10), LEFT)).toBeCloseTo(16);
+    expect(exitDistance(circle(0, 20, 10), box(0, 0, 100, 100), LEFT)).toBeCloseTo(60); // flat side
+    expect(exitDistance(circle(0, 56, 10), box(0, 0, 100, 100), LEFT)).toBeCloseTo(50 + 8); // corner: dy=6 → √(100−36)
+    expect(exitDistance(circle(0, 70, 10), box(0, 0, 100, 100), LEFT)).toBe(0);
+    // Moving away diagonally clears a corner sooner than along one axis would.
+    expect(exitDistance(box(80, 80, 100, 100), box(0, 0, 100, 100), { x: 1, y: 1 })).toBeCloseTo(
+      20 * Math.SQRT2,
+    );
+    // A wall turned by 45°, a box moving left into its slanted face: level with the corner it must
+    // back off to the corner (70.7 + 10 − 60), higher up the face is only half a unit deep along
+    // its normal – an upright frame would still demand the 20.7.
+    expect(exitDistance(box(-60, 0, 20, 20), box(0, 0, 100, 100, 45), LEFT)).toBeCloseTo(
+      50 * Math.SQRT2 + 10 - 60,
+    );
+    expect(exitDistance(box(-60, 30, 20, 20), box(0, 0, 100, 100, 45), LEFT)).toBeCloseTo(
+      (50 + 10 * Math.SQRT2 - 90 * Math.SQRT1_2) * Math.SQRT2,
+    );
   });
 });
 
diff --git a/tests/core/commands.test.ts b/tests/core/commands.test.ts
index a91476c8..9d64c7de 100644
--- a/tests/core/commands.test.ts
+++ b/tests/core/commands.test.ts
@@ -24,6 +24,10 @@ describe('asset commands', () => {
     expect(project.activeSceneId).toBe(sceneId);
     expect(findAsset(project, spriteId)?.props).toEqual({
       enabled: true,
+      scale: 1,
+      rotation: 0,
+      mirrorX: false,
+      mirrorY: false,
       width: 100,
       height: 100,
       color: 0xffffff,
diff --git a/tests/core/hierarchy.test.ts b/tests/core/hierarchy.test.ts
index e338c397..9d287dea 100644
--- a/tests/core/hierarchy.test.ts
+++ b/tests/core/hierarchy.test.ts
@@ -11,12 +11,12 @@ import {
   ancestorIdsOf,
   descendantsOf,
   findObject,
-  localPositionFor,
   type ObjectTree,
   treeOrder,
   visibleTreeOrder,
 } from '../../src/core/model/hierarchy';
 import { findScene } from '../../src/core/model/Project';
+import { localPositionFor } from '../../src/core/properties/transform';
 import { deserializeProject, serializeProject } from '../../src/core/store/serialize';
 import {
   fakeContext,
@@ -39,7 +39,7 @@ function threeBoxes() {
 
 describe('scene hierarchy', () => {
   it('re-parenting keeps the world position and moves children along with the parent', () => {
-    const { stack, sceneId, a, b, c, sceneOf, object, world } = threeBoxes();
+    const { project, stack, sceneId, a, b, c, sceneOf, object, world } = threeBoxes();
     stack.execute(new SetParentCommand(sceneId, b, a));
     // b stays at world (10,0): local becomes (-90,-100,-1).
     expect(object(b).parentId).toBe(a);
@@ -63,7 +63,7 @@ describe('scene hierarchy', () => {
     expect(object(b).parentId).toBe('');
     expect(object(b).position).toEqual({ x: 10, y: 0, z: 0 });
 
-    expect(localPositionFor(sceneOf(), a, { x: 100, y: 100, z: 1 })).toEqual({ x: 0, y: 0, z: 0 });
+    expect(localPositionFor(project, sceneOf(), a, { x: 100, y: 100, z: 1 })).toEqual({ x: 0, y: 0, z: 0 });
   });
 
   it('refuses cycles and self-parenting', () => {
diff --git a/tests/core/resolve.test.ts b/tests/core/resolve.test.ts
index 99ee7985..46a7e304 100644
--- a/tests/core/resolve.test.ts
+++ b/tests/core/resolve.test.ts
@@ -25,6 +25,10 @@ describe('PropertySchema', () => {
   it('provides defaults for every key', () => {
     expect(defaultsFor('sprite')).toEqual({
       enabled: true,
+      scale: 1,
+      rotation: 0,
+      mirrorX: false,
+      mirrorY: false,
       width: 100,
       height: 100,
       color: 0xffffff,
@@ -51,6 +55,10 @@ describe('resolve', () => {
     sprite.props = { width: 50, bogus: 1 };
     expect(resolveAssetProps(project, sprite)).toEqual({
       enabled: true,
+      scale: 1,
+      rotation: 0,
+      mirrorX: false,
+      mirrorY: false,
       width: 50,
       height: 100,
       color: 0xffffff,
@@ -65,6 +73,10 @@ describe('resolve', () => {
     const object = createSceneObject('o1', sprite.id, { x: 10, y: 20 });
     expect(resolveSceneObjectProps(project, sprite, object)).toEqual({
       enabled: true,
+      scale: 1,
+      rotation: 0,
+      mirrorX: false,
+      mirrorY: false,
       width: 100,
       height: 100,
       color: 0x00ff00,
diff --git a/tests/core/serialize.test.ts b/tests/core/serialize.test.ts
index e5718e8b..45f2fc5a 100644
--- a/tests/core/serialize.test.ts
+++ b/tests/core/serialize.test.ts
@@ -61,6 +61,10 @@ describe('serialize', () => {
     expect(project.assets[1]!.name).toBe('sprite');
     expect(project.assets[1]!.props).toEqual({
       enabled: true,
+      scale: 1,
+      rotation: 0,
+      mirrorX: false,
+      mirrorY: false,
       width: 10,
       height: 100,
       color: 0xffffff,
diff --git a/tests/core/shaders.test.ts b/tests/core/shaders.test.ts
index d6586519..8310ab5e 100644
--- a/tests/core/shaders.test.ts
+++ b/tests/core/shaders.test.ts
@@ -458,7 +458,18 @@ describe('a sprite wearing a shader', () => {
   it('has the shader picker and, once one is picked, that shader’s rows', () => {
     const f = wearing();
     const keys = () => schemaForAsset(f.project, f.sprite()).map((d) => d.key);
-    expect(keys()).toEqual(['enabled', 'width', 'height', 'color', 'texture', 'shader']);
+    expect(keys()).toEqual([
+      'enabled',
+      'scale',
+      'rotation',
+      'mirrorX',
+      'mirrorY',
+      'width',
+      'height',
+      'color',
+      'texture',
+      'shader',
+    ]);
     expect(schemaFor('sprite').find((d) => d.key === SHADER_KEY)).toMatchObject({
       kind: 'asset',
       assetTypes: ['shader'],
@@ -467,6 +478,10 @@ describe('a sprite wearing a shader', () => {
     f.stack.execute(new SetAssetPropertyCommand(f.spriteId, SHADER_KEY, f.s.id));
     expect(keys()).toEqual([
       'enabled',
+      'scale',
+      'rotation',
+      'mirrorX',
+      'mirrorY',
       'width',
       'height',
       'color',
@@ -479,14 +494,14 @@ describe('a sprite wearing a shader', () => {
     expect(assetDeviates(f.project, f.sprite())).toBe(true);
     // A dangling id adds no rows and draws nothing – the picker still shows it.
     f.sprite().props[SHADER_KEY] = 'gone';
-    expect(keys()).toHaveLength(6);
+    expect(keys()).toHaveLength(10);
   });
 
   it('gives a placed object the rows of the shader it overrides, filled with the node defaults', () => {
     const f = wearing();
     const id = place(f, f.spriteId, { x: 0, y: 0 });
     const object = () => f.object(id);
-    expect(schemaForObject(f.project, f.sprite(), object())).toHaveLength(6);
+    expect(schemaForObject(f.project, f.sprite(), object())).toHaveLength(10);
     f.stack.execute(new SetOverrideCommand(f.sceneId, id, SHADER_KEY, f.s.id));
     expect(schemaForObject(f.project, f.sprite(), object()).map((d) => d.key)).toContain('saturation');
     expect(resolveSceneObjectProps(f.project, f.sprite(), object())).toMatchObject({
diff --git a/tests/core/texture-commands.test.ts b/tests/core/texture-commands.test.ts
index 200a35dc..2c8ded4b 100644
--- a/tests/core/texture-commands.test.ts
+++ b/tests/core/texture-commands.test.ts
@@ -50,6 +50,10 @@ describe('CreateAssetCommand with starting props', () => {
 
     expect(findAsset(project, sprite.asset.id)?.props).toEqual({
       enabled: true,
+      scale: 1,
+      rotation: 0,
+      mirrorX: false,
+      mirrorY: false,
       width: 64,
       height: 32,
       color: 0xffffff,
diff --git a/tests/core/transform.test.ts b/tests/core/transform.test.ts
new file mode 100644
index 00000000..d8a728ec
--- /dev/null
+++ b/tests/core/transform.test.ts
@@ -0,0 +1,184 @@
+import { describe, expect, it } from 'vitest';
+import { CreateContainerCommand } from '../../src/core/commands/containerCommands';
+import {
+  MoveSceneObjectCommand,
+  SetOverrideCommand,
+  SetParentCommand,
+} from '../../src/core/commands/sceneObjectCommands';
+import { worldPositionOf, worldTransformOf } from '../../src/core/graph';
+import type { ContainerAsset } from '../../src/core/model/Asset';
+import {
+  apply,
+  decompose,
+  invert,
+  multiply,
+  scaleOf,
+  transformMatrix,
+  translation,
+} from '../../src/core/model/matrix';
+import { findAsset } from '../../src/core/model/Project';
+import { footprintContains, footprintCorners, footprintExtent } from '../../src/core/properties/bounds';
+import {
+  hasTransformProps,
+  isCommonProp,
+  isValidVariableName,
+  schemaFor,
+} from '../../src/core/properties/PropertySchema';
+import { carriedTransform, localPositionFor, worldTransform } from '../../src/core/properties/transform';
+import { fakeHandle, makeSpriteScene, place, startWorld } from '../helpers';
+
+const close = (p: { x: number; y: number }, x: number, y: number) => {
+  expect(p.x).toBeCloseTo(x, 9);
+  expect(p.y).toBeCloseTo(y, 9);
+};
+
+describe('matrix', () => {
+  it('turns clockwise on screen, scales about the origin, mirrors before turning', () => {
+    close(apply(transformMatrix(0, 0, 90, 1, false, false), { x: 1, y: 0 }), 0, 1);
+    close(apply(transformMatrix(10, 20, 0, 2, false, false), { x: 1, y: 1 }), 12, 22);
+    close(apply(transformMatrix(0, 0, 0, 1, true, false), { x: 1, y: 2 }), -1, 2);
+    close(apply(transformMatrix(0, 0, 90, 1, true, false), { x: 1, y: 0 }), 0, -1);
+  });
+
+  it('a product undoes through its inverse, and a similarity decomposes back into its rows', () => {
+    const m = multiply(
+      transformMatrix(5, 6, 30, 2, false, false),
+      transformMatrix(1, 2, 15, 0.5, true, false),
+    );
+    close(apply(invert(m), apply(m, { x: 3, y: 4 })), 3, 4);
+    expect(scaleOf(m)).toBeCloseTo(1);
+    expect(decompose(m)).toMatchObject({ mirrorX: true });
+    expect(decompose(m).rotation).toBeCloseTo(45);
+    const plain = decompose(transformMatrix(0, 0, 30, 2, false, false));
+    expect(plain.rotation).toBeCloseTo(30, 9);
+    expect(plain.scale).toBeCloseTo(2, 9);
+    expect(plain.mirrorX).toBe(false);
+  });
+});
+
+describe('transform chain', () => {
+  it('a child sits where its turned, sized parent puts it, in the model and in the world', () => {
+    const f = makeSpriteScene();
+    const parent = place(f, f.spriteId, { x: 100, y: 100 });
+    // `place` takes a world position; the local one is set outright.
+    const child = place(f, f.spriteId, { x: 0, y: 0 }, parent);
+    f.stack.execute(new MoveSceneObjectCommand(f.sceneId, child, { x: 50, y: 0, z: 0 }));
+    f.stack.execute(new SetOverrideCommand(f.sceneId, parent, 'rotation', 90));
+    f.stack.execute(new SetOverrideCommand(f.sceneId, parent, 'scale', 2));
+    // Local (50, 0) → scaled (100, 0) → turned a quarter clockwise (0, 100) → at (100, 200).
+    close(f.world(child), 100, 200);
+    const chain = worldTransform(f.project, f.sceneOf(), f.object(child));
+    expect(scaleOf(chain.m)).toBeCloseTo(2);
+    expect(decompose(chain.m).rotation).toBeCloseTo(90);
+
+    const { world } = startWorld(f);
+    close(worldPositionOf(world.findByObjectId(child)!), 100, 200);
+    // Behaviors write the rows like any property: the next read of the chain sees it.
+    world.findByObjectId(parent)!.props.rotation = 0;
+    close(worldPositionOf(world.findByObjectId(child)!), 200, 100);
+  });
+
+  it('the rows are reserved names shared by drawn types, absent from cameras and behaviors', () => {
+    expect(schemaFor('sprite').map((d) => d.key)).toEqual(
+      expect.arrayContaining(['scale', 'rotation', 'mirrorX', 'mirrorY']),
+    );
+    expect(hasTransformProps('camera')).toBe(false);
+    expect(schemaFor('camera').some((d) => d.key === 'rotation')).toBe(false);
+    expect(isCommonProp('rotation')).toBe(true);
+    expect(isValidVariableName('scale')).toBe(false);
+    // A camera under a turned parent is still placed by the chain; its own props carry no rows.
+    const parent = fakeHandle({ rotation: 90 });
+    const camera = { ...fakeHandle({}, parent), x: 10, y: 0 };
+    close(worldPositionOf(camera), 0, 10);
+    expect(scaleOf(worldTransformOf(camera).m)).toBeCloseTo(1);
+  });
+
+  it('the local position for a world point goes back through the parent transform', () => {
+    const f = makeSpriteScene();
+    const parent = place(f, f.spriteId, { x: 100, y: 100 });
+    f.stack.execute(new SetOverrideCommand(f.sceneId, parent, 'rotation', 90));
+    const local = localPositionFor(f.project, f.sceneOf(), parent, { x: 100, y: 150, z: 0 });
+    close(local, 50, 0);
+    const child = place(f, f.spriteId, { x: 100, y: 150 }, parent);
+    close(f.world(child), 100, 150);
+  });
+});
+
+describe('carried transform', () => {
+  it('re-parenting keeps the look: rows absorb the parent chain, nothing is written between upright parents', () => {
+    const f = makeSpriteScene();
+    const turned = place(f, f.spriteId, { x: 0, y: 0 });
+    const upright = place(f, f.spriteId, { x: 300, y: 0 });
+    const box = place(f, f.spriteId, { x: 100, y: 0 });
+    f.stack.execute(new SetOverrideCommand(f.sceneId, turned, 'rotation', 90));
+    f.stack.execute(new SetOverrideCommand(f.sceneId, turned, 'scale', 2));
+    f.stack.execute(new SetOverrideCommand(f.sceneId, box, 'mirrorY', true));
+
+    f.stack.execute(new SetParentCommand(f.sceneId, box, upright));
+    expect(f.object(box).overrides).toEqual({ mirrorY: true });
+    close(f.world(box), 100, 0);
+
+    f.stack.execute(new SetParentCommand(f.sceneId, box, turned));
+    close(f.world(box), 100, 0);
+    const world = worldTransform(f.project, f.sceneOf(), f.object(box)).m;
+    expect(decompose(world)).toMatchObject({ mirrorX: true });
+    expect(scaleOf(world)).toBeCloseTo(1);
+    expect(f.object(box).overrides.rotation).toBeCloseTo(-90);
+    expect(f.object(box).overrides.scale).toBeCloseTo(0.5);
+    expect(f.object(box).overrides.mirrorY).toBe(true);
+
+    f.stack.undo();
+    expect(f.object(box).overrides).toEqual({ mirrorY: true });
+    expect(f.object(box).parentId).toBe(upright);
+  });
+
+  it('a mirror in the chain is absorbed as a mirror, whichever way it was written', () => {
+    const own = { rotation: 0, scale: 1, mirrorX: false, mirrorY: false };
+    const before = transformMatrix(0, 0, 0, 1, true, false);
+    const carried = carriedTransform(own, before, translation(0, 0));
+    expect(carried).toMatchObject({ mirrorX: true, mirrorY: false });
+    expect(carried.scale).toBeCloseTo(1);
+    expect(carried.rotation).toBeCloseTo(0);
+  });
+
+  it('grouping under a turned parent keeps every root where it was, rows included', () => {
+    const f = makeSpriteScene();
+    const parent = place(f, f.spriteId, { x: 0, y: 0 });
+    const a = place(f, f.spriteId, { x: 100, y: 0 }, parent);
+    const b = place(f, f.spriteId, { x: 100, y: 50 }, parent);
+    f.stack.execute(new SetOverrideCommand(f.sceneId, parent, 'rotation', 90));
+    const worldsBefore = [f.world(a), f.world(b)];
+    const create = new CreateContainerCommand(f.sceneId, [a, b]);
+    f.stack.execute(create);
+    const asset = findAsset(f.project, create.assetId) as ContainerAsset;
+    // Same parent for both: positions relative to the container in the parent's frame, no rows.
+    expect(asset.objects.map((o) => o.overrides)).toEqual([{}, {}]);
+    const contents = f.viewOf().objects.filter((o) => o.parentId === create.objectId);
+    expect(contents).toHaveLength(2);
+    const worldsAfter = contents.map((o) => f.world(o.id));
+    for (const before of worldsBefore)
+      expect(
+        worldsAfter.some((w) => Math.abs(w.x - before.x) < 1e-9 && Math.abs(w.y - before.y) < 1e-9),
+      ).toBe(true);
+  });
+});
+
+describe('footprint under a transform', () => {
+  it('corners, extent and hit test follow the turn, symbols stay upright', () => {
+    const m = transformMatrix(100, 100, 90, 2, false, false);
+    const props = { width: 50, height: 10 };
+    const extent = footprintExtent('sprite', props, m);
+    expect(extent.left).toBeCloseTo(90);
+    expect(extent.right).toBeCloseTo(110);
+    expect(extent.top).toBeCloseTo(50);
+    expect(extent.bottom).toBeCloseTo(150);
+    expect(footprintContains('sprite', props, m, { x: 100, y: 145 })).toBe(true);
+    expect(footprintContains('sprite', props, m, { x: 145, y: 100 })).toBe(false);
+    // Two world units of padding around a box drawn at scale 2 is one local unit on every side.
+    const padded = footprintCorners('sprite', props, m, 2);
+    expect(padded[0]!.y).toBeCloseTo(48);
+    // A behavior's diamond keeps its size and stays upright: the frame does not scale by 2.
+    const symbol = footprintExtent('behavior', {}, m);
+    expect(symbol.right - symbol.left).toBe(28);
+  });
+});
diff --git a/tests/helpers.ts b/tests/helpers.ts
index 790a6203..277fd157 100644
--- a/tests/helpers.ts
+++ b/tests/helpers.ts
@@ -19,7 +19,8 @@ import {
   VARIABLES,
 } from '../src/core/graph';
 import type { BehaviorAsset, SceneAsset } from '../src/core/model/Asset';
-import { findObject, worldPosition } from '../src/core/model/hierarchy';
+import { findObject } from '../src/core/model/hierarchy';
+import { worldPosition } from '../src/core/properties/transform';
 import { resolvedHolder } from '../src/core/model/instance';
 import type { Position, PropertyBag, Vec2 } from '../src/core/model/Properties';
 import { createEmptyProject, findScene, type Project } from '../src/core/model/Project';
@@ -70,7 +71,7 @@ export function makeSpriteScene(spriteName = 'Box'): SceneFixture {
     sceneOf,
     viewOf,
     object,
-    world: (id) => worldPosition(viewOf(), object(id)),
+    world: (id) => worldPosition(fixture.project, viewOf(), object(id)),
   };
 }
 
diff --git a/tools/e2e/28-shader.mjs b/tools/e2e/28-shader.mjs
index 177124f2..0cbbdee5 100644
--- a/tools/e2e/28-shader.mjs
+++ b/tools/e2e/28-shader.mjs
@@ -1,4 +1,5 @@
 import { colourDistance } from '../lib/harness.mjs';
+import { FIND_HALVES, HALVES_LEFT as LEFT, HALVES_RIGHT as RIGHT, PASTE_HALVES } from './lib/pictures.mjs';
 
 /**
  * Shader assets: made from the + menu, built from nodes in the Shader tab, worn by a sprite through
@@ -10,22 +11,6 @@ import { colourDistance } from '../lib/harness.mjs';
 const BASE = '#4080ff';
 const ADJUST = 'adjust-colors';
 
-/** A 32 px picture, red on the left and green on the right, pasted in for the sampling checks (28l). */
-const PASTE_HALVES = `(async () => {
-  const canvas = document.createElement('canvas');
-  canvas.width = 32;
-  canvas.height = 32;
-  const g = canvas.getContext('2d');
-  g.fillStyle = '#e63c3c'; g.fillRect(0, 0, 16, 32);
-  g.fillStyle = '#3cc85a'; g.fillRect(16, 0, 16, 32);
-  const blob = await new Promise((resolve) => canvas.toBlob(resolve, 'image/png'));
-  const transfer = new DataTransfer();
-  transfer.items.add(new File([blob], 'halves.png', { type: 'image/png' }));
-  window.dispatchEvent(new ClipboardEvent('paste', { clipboardData: transfer, bubbles: true }));
-})()`;
-const LEFT = '#e63c3c';
-const RIGHT = '#3cc85a';
-
 export async function shader(t) {
   const {
     check,
@@ -457,11 +442,7 @@ export async function shader(t) {
   // with the far half, Mirror with the near one.
   await evaluate(PASTE_HALVES);
   await sleep(2500);
-  const picture = (
-    await json(
-      "__gm.state.project.assets.filter((a) => a.type === 'texture' && a.width === 32 && a.height === 32).map((a) => ({ id: a.id, file: a.file }))",
-    )
-  )[0];
+  const picture = await json(FIND_HALVES);
   check(
     'a two-colour picture is pasted in for the sampling checks',
     Boolean(picture),
diff --git a/tools/e2e/29-transform.mjs b/tools/e2e/29-transform.mjs
new file mode 100644
index 00000000..958ddb3c
--- /dev/null
+++ b/tools/e2e/29-transform.mjs
@@ -0,0 +1,266 @@
+import { colourDistance } from '../lib/harness.mjs';
+import { FIND_HALVES, HALVES_LEFT, HALVES_RIGHT, PASTE_HALVES } from './lib/pictures.mjs';
+import { LANES } from './lib/scaffold.mjs';
+
+/**
+ * Scale, rotation and mirror (#76): rows of every drawn object, cascading down the hierarchy like
+ * a position. Read in the pixels of the scene view, in where a child lands under a turned parent,
+ * in a drag that goes back through the parent's frame, and in a turned collider stopping a mover.
+ * Everything lives in its own lane, looked at through `__gm.lookAt`: the origin is crowded by
+ * what the earlier steps left, and symbols draw above objects.
+ */
+const RED = '#ff0000';
+const SPRITE = 'tf-sprite';
+const BOX = 'otf';
+const CHILD = 'otf-child';
+const CAMERA = 'otf-cam';
+const RIG = ['otf-wall', 'otf-wallc', 'otf-body', 'otf-bodyc', 'otf-logic'];
+const LANE = LANES.transform;
+
+export async function transform(t) {
+  const {
+    check,
+    sleep,
+    json,
+    evaluate,
+    mutate,
+    select,
+    drag,
+    undo,
+    press,
+    holdKey,
+    object,
+    entity,
+    stop,
+    shot,
+  } = t;
+  const field = (label) => json(`__gm.inspectorField(${JSON.stringify(label)})`);
+  const rect = (id) => json(`__gm.objectRect(${JSON.stringify(id)})`);
+  const screen = (id) => json(`__gm.objectScreen(${JSON.stringify(id)})`);
+  const override = (id, patch) =>
+    mutate(
+      `Object.assign(__gm.state.activeScene.objects.find((o) => o.id === ${JSON.stringify(id)}).overrides, ${JSON.stringify(patch)})`,
+    );
+  const spriteProps = (patch) =>
+    mutate(
+      `Object.assign(__gm.state.project.assets.find((a) => a.id === ${JSON.stringify(SPRITE)}).props, ${JSON.stringify(patch)})`,
+    );
+  const selectObject = async (id) => {
+    await select({ kind: 'object', holderId: 'scene1', objectIds: [id], objectId: id });
+    await sleep(300);
+  };
+  const remove = (ids) =>
+    mutate(`
+      const gone = new Set(${JSON.stringify(ids)});
+      const scene = __gm.state.activeScene;
+      scene.objects = scene.objects.filter((o) => !gone.has(o.id))`);
+
+  await press('Escape');
+  await evaluate('__gm.state.select(null)');
+  await evaluate("__gm.showTab('scene')");
+  await sleep(200);
+
+  // 29a. A plain red 100×40 sprite in its lane, and a second camera to read the inspector off;
+  //      a sprite object has the transform rows, a camera object has not.
+  await mutate(`
+    const project = __gm.state.project;
+    const sprite = structuredClone(project.assets.find((a) => a.id === 'crate'));
+    sprite.id = ${JSON.stringify(SPRITE)};
+    sprite.name = 'Turned';
+    sprite.props = { ...sprite.props, width: 100, height: 40, color: 0xff0000, texture: '', shader: '' };
+    project.assets.push(sprite);
+    const scene = __gm.state.activeScene;
+    const make = (id, assetId, parentId, position, overrides = {}) => ({ id, assetId, name: '', position, editorOffset: { x: 0, y: 0 }, overrides, parentId, instanceEdits: {} });
+    scene.objects.push(
+      make(${JSON.stringify(BOX)}, ${JSON.stringify(SPRITE)}, '', { x: 0, y: ${LANE}, z: 960 }),
+      make(${JSON.stringify(CAMERA)}, 'cam', '', { x: 0, y: ${LANE - 400}, z: 0 }, { enabled: false }),
+    );
+    __gm.lookAt(0, ${LANE});
+    __gm.state.select(null)`);
+  await sleep(500);
+  await selectObject(BOX);
+  const rows = {
+    title: await t.has('TRANSFORM'),
+    scale: await field('Scale'),
+    rotation: await field('Rotation'),
+    mirrorX: await field('Mirror X'),
+    mirrorY: await field('Mirror Y'),
+  };
+  check(
+    'a sprite object has a TRANSFORM section with Scale, Rotation, Mirror X and Mirror Y',
+    rows.title && rows.scale && rows.rotation && rows.mirrorX && rows.mirrorY,
+    JSON.stringify(rows),
+  );
+  await selectObject(CAMERA);
+  check(
+    'a camera object has no transform rows',
+    (await field('Rotation')) === null && (await field('Scale')) === null && !(await t.has('TRANSFORM')),
+  );
+  await selectObject(BOX);
+
+  // 29b. Pixels: world units from the sprite's centre, at the view's scale.
+  const base = await rect(BOX);
+  check(
+    'the sprite is on screen',
+    Boolean(base) && base.w > 8,
+    JSON.stringify({ base, viewport: await json('__gm.viewportRect()') }),
+  );
+  if (!base) return;
+  const s = base.w / 100;
+  const px = (dx, dy) => json(`__gm.pixelAt(${Math.round(base.x + dx * s)}, ${Math.round(base.y + dy * s)})`);
+  const isRed = (c) => colourDistance(c, RED) < 12;
+  const upright = { right: await px(40, 0), below: await px(0, 40) };
+  check(
+    'upright: red 40 to the right of the centre, not red 40 below (the box is 40 tall)',
+    isRed(upright.right) && !isRed(upright.below),
+    JSON.stringify(upright),
+  );
+
+  // 29c. Rotation 90: the box stands up – its frame swaps sides, the pixels move with it.
+  await override(BOX, { rotation: 90 });
+  await sleep(300);
+  const turned = await rect(BOX);
+  const turnedPx = { right: await px(40, 0), below: await px(0, 40) };
+  check(
+    'rotation 90 turns the box: frame 40 wide and 100 tall, red below the centre, not to the right',
+    Math.abs(turned.w - 40 * s) < 2 &&
+      Math.abs(turned.h - 100 * s) < 2 &&
+      isRed(turnedPx.below) &&
+      !isRed(turnedPx.right),
+    JSON.stringify({ turned, turnedPx }),
+  );
+
+  // 29d. Scale 2 on top: twice the frame, red where the upright box would have ended long ago.
+  await override(BOX, { scale: 2 });
+  await sleep(300);
+  const grown = await rect(BOX);
+  const far = await px(0, 90);
+  check(
+    'scale 2 doubles the turned box: frame 80 by 200, red 90 below the centre',
+    Math.abs(grown.w - 80 * s) < 2 && Math.abs(grown.h - 200 * s) < 2 && isRed(far),
+    JSON.stringify({ grown, far }),
+  );
+  await shot('29-turned-scaled');
+
+  // 29e. Mirror, read on a two-colour picture: red left, green right; Mirror X swaps them, Mirror Y
+  //      does not (the halves are side by side).
+  await override(BOX, { rotation: 0, scale: 1 });
+  await evaluate(PASTE_HALVES);
+  await sleep(2500);
+  const picture = await json(FIND_HALVES);
+  check('a two-colour picture is pasted in for the mirror checks', Boolean(picture), JSON.stringify(picture));
+  await evaluate('__gm.state.select(null)');
+  await spriteProps({ texture: picture?.id ?? '', color: 0xffffff });
+  await sleep(600);
+  const plain = { left: await px(-30, 0), right: await px(30, 0) };
+  check(
+    'the picture is on: red left, green right',
+    colourDistance(plain.left, HALVES_LEFT) < 12 && colourDistance(plain.right, HALVES_RIGHT) < 12,
+    JSON.stringify(plain),
+  );
+  await override(BOX, { mirrorX: true });
+  await sleep(300);
+  const mirrored = { left: await px(-30, 0), right: await px(30, 0) };
+  check(
+    'Mirror X swaps the halves: green left, red right',
+    colourDistance(mirrored.left, HALVES_RIGHT) < 12 && colourDistance(mirrored.right, HALVES_LEFT) < 12,
+    JSON.stringify(mirrored),
+  );
+  await override(BOX, { mirrorY: true });
+  await sleep(300);
+  const both = { left: await px(-30, 0), right: await px(30, 0) };
+  check(
+    'Mirror Y on top leaves the halves where Mirror X put them',
+    colourDistance(both.left, HALVES_RIGHT) < 12 && colourDistance(both.right, HALVES_LEFT) < 12,
+    JSON.stringify(both),
+  );
+  await override(BOX, { mirrorX: false, mirrorY: false });
+  await spriteProps({ texture: '', color: 0xff0000 });
+  await sleep(300);
+
+  // 29f. A child 100 to the right of its parent: once the parent turns a quarter, the child hangs
+  //      100 below it – and is drawn turned itself.
+  await mutate(`
+    const scene = __gm.state.activeScene;
+    scene.objects.push({ id: ${JSON.stringify(CHILD)}, assetId: ${JSON.stringify(SPRITE)}, name: '', position: { x: 100, y: 0, z: 1 }, editorOffset: { x: 0, y: 0 }, overrides: {}, parentId: ${JSON.stringify(BOX)}, instanceEdits: {} });`);
+  await sleep(300);
+  const flat = { parent: await screen(BOX), child: await screen(CHILD) };
+  await override(BOX, { rotation: 90 });
+  await sleep(300);
+  const hung = { parent: await screen(BOX), child: await screen(CHILD), childRect: await rect(CHILD) };
+  check(
+    'a child follows its parent’s turn: 100 to the right becomes 100 below, drawn turned',
+    Math.abs(flat.child.x - (flat.parent.x + 100 * s)) < 2 &&
+      Math.abs(flat.child.y - flat.parent.y) < 2 &&
+      Math.abs(hung.child.x - hung.parent.x) < 2 &&
+      Math.abs(hung.child.y - (hung.parent.y + 100 * s)) < 2 &&
+      Math.abs(hung.childRect.w - 40 * s) < 2 &&
+      Math.abs(hung.childRect.h - 100 * s) < 2,
+    JSON.stringify({ flat, hung, s }),
+  );
+
+  // 29g. Dragging the child 30 screen pixels down moves it along the parent's x axis – which now
+  //      points down – so its local x grows by 30 / scale and y stays.
+  await drag(hung.child.x, hung.child.y, hung.child.x, hung.child.y + 30, { steps: 8 });
+  await sleep(300);
+  const dragged = await object(CHILD);
+  check(
+    'a drag under a turned parent lands in the parent’s frame: local x + 30 / scale, y unchanged',
+    Math.abs(dragged.position.x - (100 + 30 / s)) <= 1 && Math.abs(dragged.position.y) <= 1,
+    JSON.stringify({ position: dragged.position, expected: 100 + 30 / s }),
+  );
+  await undo();
+  await sleep(200);
+  check('undo puts the child back', (await object(CHILD)).position.x === 100);
+  await selectObject(CHILD);
+  const gizmos = await json('__gm.gizmos()');
+  check(
+    'the turned child is outlined like any selection',
+    gizmos.some((g) => g.id === CHILD && g.primary && g.alpha === 1),
+    JSON.stringify(gizmos),
+  );
+  await shot('29-turned-child');
+
+  // 29h. In the game a turned collider blocks where it really is. A crate body with the fixture's
+  //      Mover walks right into a wall whose 140-wide collider stands on end (60 wide): it parks
+  //      flush at −80 – upright, the wall would stop it at −120 (see 8c).
+  await mutate(`
+    const scene = __gm.state.activeScene;
+    const make = (id, assetId, parentId, position, overrides = {}) => ({ id, assetId, name: '', position, editorOffset: { x: 0, y: 0 }, overrides, parentId, instanceEdits: {} });
+    scene.objects.push(
+      make('otf-wall', 'crate', '', { x: 0, y: ${LANE + 300}, z: 0 }, { rotation: 90 }),
+      make('otf-wallc', 'collider', 'otf-wall', { x: 0, y: 0, z: 0 }, { width: 140 }),
+      make('otf-body', 'crate', '', { x: -200, y: ${LANE + 300}, z: 0 }),
+      make('otf-bodyc', 'collider', 'otf-body', { x: 0, y: 0, z: 0 }),
+      make('otf-logic', 'mover', 'otf-body', { x: 0, y: 0, z: 0 }),
+    );
+    __gm.state.select(null)`);
+  await press('Enter', { ctrl: true });
+  await sleep(300);
+  await holdKey('ArrowRight', 1000);
+  const body = await entity('otf-body');
+  await stop();
+  check(
+    'a turned collider stops the mover where the turned box is: flush at x = −80',
+    body && Math.abs(body.x - -80) < 0.5,
+    `world x=${body?.x}`,
+  );
+  await remove(RIG);
+
+  // 29i. Cleanup, and the view back where the earlier steps left the scene.
+  await remove([BOX, CHILD, CAMERA]);
+  await mutate(`
+    const project = __gm.state.project;
+    const gone = new Set([${JSON.stringify(SPRITE)}, ${JSON.stringify(picture?.id ?? '')}]);
+    project.assets = project.assets.filter((a) => !gone.has(a.id));
+    __gm.lookAt(0, 0)`);
+  await evaluate('__gm.state.select(null)');
+  await sleep(200);
+  check(
+    'the step leaves nothing behind',
+    (await json(`__gm.state.project.assets.some((a) => a.id === ${JSON.stringify(SPRITE)})`)) === false &&
+      (await json(`__gm.state.activeScene.objects.some((o) => o.id.startsWith('otf'))`)) === false &&
+      (await json(`__gm.state.project.assets.some((a) => a.id === ${JSON.stringify(picture?.id ?? '')})`)) ===
+        false,
+  );
+}
diff --git a/tools/e2e/index.mjs b/tools/e2e/index.mjs
index b57af495..eb2a54d7 100644
--- a/tools/e2e/index.mjs
+++ b/tools/e2e/index.mjs
@@ -31,6 +31,7 @@ import { animation } from './25-animation.mjs';
 import { stateMachine } from './26-state-machine.mjs';
 import { consolePanel } from './27-console.mjs';
 import { shader } from './28-shader.mjs';
+import { transform } from './29-transform.mjs';
 
 export const STEPS = [
   hierarchy,
@@ -61,4 +62,5 @@ export const STEPS = [
   stateMachine,
   consolePanel,
   shader,
+  transform,
 ];
diff --git a/tools/e2e/lib/pictures.mjs b/tools/e2e/lib/pictures.mjs
new file mode 100644
index 00000000..0e782dc2
--- /dev/null
+++ b/tools/e2e/lib/pictures.mjs
@@ -0,0 +1,19 @@
+/** A 32 px picture, red on the left and green on the right, pasted in for pixel checks (28l, 29e). */
+export const PASTE_HALVES = `(async () => {
+  const canvas = document.createElement('canvas');
+  canvas.width = 32;
+  canvas.height = 32;
+  const g = canvas.getContext('2d');
+  g.fillStyle = '#e63c3c'; g.fillRect(0, 0, 16, 32);
+  g.fillStyle = '#3cc85a'; g.fillRect(16, 0, 16, 32);
+  const blob = await new Promise((resolve) => canvas.toBlob(resolve, 'image/png'));
+  const transfer = new DataTransfer();
+  transfer.items.add(new File([blob], 'halves.png', { type: 'image/png' }));
+  window.dispatchEvent(new ClipboardEvent('paste', { clipboardData: transfer, bubbles: true }));
+})()`;
+export const HALVES_LEFT = '#e63c3c';
+export const HALVES_RIGHT = '#3cc85a';
+
+/** The pasted picture's asset, once the paste has landed. */
+export const FIND_HALVES =
+  "__gm.state.project.assets.filter((a) => a.type === 'texture' && a.width === 32 && a.height === 32).map((a) => ({ id: a.id, file: a.file }))[0] ?? null";
diff --git a/tools/e2e/lib/scaffold.mjs b/tools/e2e/lib/scaffold.mjs
index ce6c66d7..a3d7af2d 100644
--- a/tools/e2e/lib/scaffold.mjs
+++ b/tools/e2e/lib/scaffold.mjs
@@ -16,6 +16,7 @@ export const LANES = {
   keyRepeat: -300,
   stateMachine: -700,
   console: -1100,
+  transform: -1500,
 };
 
 /**