diff --git a/doc/whats-new.rst b/doc/whats-new.rst
index add40bb6b81..39c6a8924f4 100644
--- a/doc/whats-new.rst
+++ b/doc/whats-new.rst
@@ -30,6 +30,9 @@ Bug fixes
 - Fix ``KeyError`` when passing a ``dim`` argument different from the default to ``convert_calendar`` (:pull:`10544`).
   By `Eric Jansen <https://github.com/ej81>`_.
 
+- Fix transpose of boolean arrays read from disk. (:issue:`10536`)
+  By `Deepak Cherian <https://github.com/dcherian>`_.
+
 
 Documentation
 ~~~~~~~~~~~~~
diff --git a/xarray/coding/variables.py b/xarray/coding/variables.py
index eff08c74500..bff0f529124 100644
--- a/xarray/coding/variables.py
+++ b/xarray/coding/variables.py
@@ -70,6 +70,9 @@ def __getitem__(self, key) -> Self:
     def get_duck_array(self):
         return duck_array_ops.astype(self.array.get_duck_array(), dtype=self.dtype)
 
+    def transpose(self, order):
+        return type(self)(self.array.transpose(order))
+
 
 class BoolTypeArray(indexing.ExplicitlyIndexedNDArrayMixin):
     """Decode arrays on the fly from integer to boolean datatype
@@ -111,6 +114,9 @@ def __getitem__(self, key) -> Self:
     def get_duck_array(self):
         return duck_array_ops.astype(self.array.get_duck_array(), dtype=self.dtype)
 
+    def transpose(self, order):
+        return type(self)(self.array.transpose(order))
+
 
 def _apply_mask(
     data: np.ndarray,
diff --git a/xarray/tests/test_backends.py b/xarray/tests/test_backends.py
index 6997be200b1..93329b2297d 100644
--- a/xarray/tests/test_backends.py
+++ b/xarray/tests/test_backends.py
@@ -308,7 +308,15 @@ def create_encoded_unsigned_false_masked_scaled_data(dtype: np.dtype) -> Dataset
 
 def create_boolean_data() -> Dataset:
     attributes = {"units": "-"}
-    return Dataset({"x": ("t", [True, False, False, True], attributes)})
+    return Dataset(
+        {
+            "x": (
+                ("t", "x"),
+                [[False, True, False, True], [True, False, False, True]],
+                attributes,
+            )
+        }
+    )
 
 
 class TestCommon:
@@ -726,6 +734,9 @@ def test_roundtrip_boolean_dtype(self) -> None:
             with self.roundtrip(actual) as actual2:
                 assert_identical(original, actual2)
                 assert actual2["x"].dtype == "bool"
+            with self.roundtrip(actual) as actual3:
+                # GH10536
+                assert_identical(original.transpose(), actual3.transpose())
 
     def test_orthogonal_indexing(self) -> None:
         in_memory = create_test_data()
diff --git a/xarray/tests/test_conventions.py b/xarray/tests/test_conventions.py
index ce792c83740..4b3729e2724 100644
--- a/xarray/tests/test_conventions.py
+++ b/xarray/tests/test_conventions.py
@@ -37,6 +37,10 @@ def test_booltype_array(self) -> None:
         assert bx.dtype == bool
         assert_array_equal(bx, np.array([True, False, True, True, False], dtype=bool))
 
+        x = np.array([[1, 0, 1], [0, 1, 0]], dtype="i1")
+        bx = coding.variables.BoolTypeArray(x)
+        assert_array_equal(bx.transpose((1, 0)), x.transpose((1, 0)))
+
 
 class TestNativeEndiannessArray:
     def test(self) -> None:
@@ -47,6 +51,11 @@ def test(self) -> None:
         assert a.dtype == expected[:].dtype
         assert_array_equal(a, expected)
 
+        y = np.arange(6, dtype=">i8").reshape((2, 3))
+        b = coding.variables.NativeEndiannessArray(y)
+        expected2 = np.arange(6, dtype="int64").reshape((2, 3))
+        assert_array_equal(b.transpose((1, 0)), expected2.transpose((1, 0)))
+
 
 def test_decode_cf_with_conflicting_fill_missing_value() -> None:
     expected = Variable(["t"], [np.nan, np.nan, 2], {"units": "foobar"})