diff --git a/masque/test/test_utils.py b/masque/test/test_utils.py
index f5e0215..fcef0de 100644
--- a/masque/test/test_utils.py
+++ b/masque/test/test_utils.py
@@ -89,6 +89,23 @@ def test_rotation_matrix_non_manhattan() -> None:
     assert_allclose(m, [[s, -s], [s, s]], atol=1e-10)
 
 
+def test_rotation_matrix_canonicalizes_before_caching() -> None:
+    expected = rotation_matrix_2d(pi / 2)
+
+    for angle in (pi / 2 + 1e-12, pi / 2 - 1e-12, -3 * pi / 2, 5 * pi / 2):
+        assert rotation_matrix_2d(angle) is expected
+
+    assert not expected.flags.writeable
+
+
+def test_rotation_matrix_does_not_snap_non_manhattan_angle() -> None:
+    cardinal = rotation_matrix_2d(pi / 2)
+    nearby = rotation_matrix_2d(pi / 2 + 2e-3)
+
+    assert nearby is not cardinal
+    assert not numpy.array_equal(nearby, cardinal)
+
+
 def test_apply_transforms() -> None:
     # cumulative [x_offset, y_offset, rotation (rad), mirror_x (0 or 1)]
     t1 = [10, 20, 0, 0]
diff --git a/masque/utils/transform.py b/masque/utils/transform.py
index 7b39122..935b0b3 100644
--- a/masque/utils/transform.py
+++ b/masque/utils/transform.py
@@ -3,6 +3,7 @@ Geometric transforms
 """
 from collections.abc import Sequence
 from functools import lru_cache
+from math import acos
 
 import numpy
 from numpy.typing import NDArray, ArrayLike
@@ -13,8 +14,26 @@ from numpy import pi
 R90 = pi / 2
 R180 = pi
 
+# Preserve the effective tolerance of the historical
+# ``isclose(cos(4 * theta), 1, atol=1e-12)`` Manhattan check. Expressing it as
+# an angular distance lets us canonicalize before consulting the matrix cache.
+_MANHATTAN_SNAP_ATOL = acos(1 - (1e-5 + 1e-12)) / 4
+_CARDINAL_ANGLES = (0.0, R90, R180, 3 * R90)
+
 
 @lru_cache
+def _rotation_matrix_2d(theta: float) -> NDArray[numpy.float64]:
+    """Build and cache an immutable matrix for a canonicalized angle."""
+    arr = numpy.array([[numpy.cos(theta), -numpy.sin(theta)],
+                       [numpy.sin(theta), +numpy.cos(theta)]])
+
+    if theta in _CARDINAL_ANGLES:
+        arr = numpy.round(arr)
+
+    arr.flags.writeable = False
+    return arr
+
+
 def rotation_matrix_2d(theta: float) -> NDArray[numpy.float64]:
     """
     2D rotation matrix for rotating counterclockwise around the origin.
@@ -25,16 +44,11 @@ def rotation_matrix_2d(theta: float) -> NDArray[numpy.float64]:
     Returns:
         rotation matrix
     """
-    arr = numpy.array([[numpy.cos(theta), -numpy.sin(theta)],
-                       [numpy.sin(theta), +numpy.cos(theta)]])
-
-    # If this was a manhattan rotation, round to remove some inaccuracies in sin & cos
-    # cos(4*theta) is 1 for any multiple of pi/2.
-    if numpy.isclose(numpy.cos(4 * theta), 1, atol=1e-12):
-        arr = numpy.round(arr)
-
-    arr.flags.writeable = False
-    return arr
+    theta = float(theta)
+    quarter_turn = round(theta / R90)
+    if abs(theta - quarter_turn * R90) <= _MANHATTAN_SNAP_ATOL:
+        theta = _CARDINAL_ANGLES[quarter_turn % 4]
+    return _rotation_matrix_2d(theta)
 
 
 def normalize_mirror(mirrored: Sequence[bool]) -> tuple[bool, float]: