diff --git a/docs/src/whatsnew/latest.rst b/docs/src/whatsnew/latest.rst
index 8b986e0ac3..baf9f4d30d 100644
--- a/docs/src/whatsnew/latest.rst
+++ b/docs/src/whatsnew/latest.rst
@@ -62,6 +62,10 @@ This document explains the changes made to Iris for this release
 #. `@gcaria`_ fixed :meth:`~iris.cube.Cube.ancillary_variable_dims` to also accept
    the string name of a :class:`~iris.coords.AncillaryVariable`. (:pull:`3931`)
 
+#. `@rcomer`_ fixed :meth:`~iris.cube.Cube.intersection` for special cases
+   where one cell's bounds align with the requested maximum and minimum, as
+   reported in :issue:`3391`. (:pull:`4046`)
+
 
 💣 Incompatible Changes
 =======================
diff --git a/lib/iris/cube.py b/lib/iris/cube.py
index a15951900b..a8c202a21f 100644
--- a/lib/iris/cube.py
+++ b/lib/iris/cube.py
@@ -3186,18 +3186,19 @@ def _intersect_modulus(
                     pre_wrap_delta != post_wrap_delta
                 )
 
-                # Recalculate the extended minimum.
                 indices = inside_indices[split_cell_indices]
                 cells = bounds[indices]
-                cells_delta = np.diff(coord.bounds[indices])
-
-                # Watch out for ascending/descending bounds
-                if cells_delta[0, 0] > 0:
-                    cells[:, 0] = cells[:, 1] - cells_delta[:, 0]
-                    minimum = np.min(cells[:, 0])
-                else:
-                    cells[:, 1] = cells[:, 0] + cells_delta[:, 0]
-                    minimum = np.min(cells[:, 1])
+                if maximum % modulus not in cells:
+                    # Recalculate the extended minimum.
+                    cells_delta = np.diff(coord.bounds[indices])
+
+                    # Watch out for ascending/descending bounds.
+                    if cells_delta[0, 0] > 0:
+                        cells[:, 0] = cells[:, 1] - cells_delta[:, 0]
+                        minimum = np.min(cells[:, 0])
+                    else:
+                        cells[:, 1] = cells[:, 0] + cells_delta[:, 0]
+                        minimum = np.min(cells[:, 1])
 
             points = wrap_lons(coord.points, minimum, modulus)
 
diff --git a/lib/iris/tests/unit/cube/test_Cube.py b/lib/iris/tests/unit/cube/test_Cube.py
index ded401cab3..3efc6a93f8 100644
--- a/lib/iris/tests/unit/cube/test_Cube.py
+++ b/lib/iris/tests/unit/cube/test_Cube.py
@@ -1731,6 +1731,14 @@ def test_aligned_exclusive(self):
         self.assertEqual(result.data[0, 0, 0], 171)
         self.assertEqual(result.data[0, 0, -1], 189)
 
+    def test_aligned_bounds_at_modulus(self):
+        cube = create_cube(-179.5, 180.5, bounds=True)
+        result = cube.intersection(longitude=(0, 360))
+        self.assertArrayEqual(result.coord("longitude").bounds[0], [0, 1])
+        self.assertArrayEqual(result.coord("longitude").bounds[-1], [359, 360])
+        self.assertEqual(result.data[0, 0, 0], 180)
+        self.assertEqual(result.data[0, 0, -1], 179)
+
     def test_negative_misaligned_points_inside(self):
         cube = create_cube(0, 360, bounds=True)
         result = cube.intersection(longitude=(-10.25, 10.25))