diff --git a/xarray/core/dataarray.py b/xarray/core/dataarray.py
index 192d73a63d3..37369afbf96 100644
--- a/xarray/core/dataarray.py
+++ b/xarray/core/dataarray.py
@@ -534,10 +534,10 @@ def _replace_maybe_drop_dims(
         variable: Variable,
         name: Hashable | None | Default = _default,
     ) -> Self:
-        if variable.dims == self.dims and variable.shape == self.shape:
+        if self.sizes == variable.sizes:
             coords = self._coords.copy()
             indexes = self._indexes
-        elif variable.dims == self.dims:
+        elif set(self.dims) == set(variable.dims):
             # Shape has changed (e.g. from reduce(..., keepdims=True)
             new_sizes = dict(zip(self.dims, variable.shape, strict=True))
             coords = {
diff --git a/xarray/tests/test_groupby.py b/xarray/tests/test_groupby.py
index 9d9c22cfa96..dc869cc3a34 100644
--- a/xarray/tests/test_groupby.py
+++ b/xarray/tests/test_groupby.py
@@ -2919,3 +2919,16 @@ def test_gappy_resample_reductions(reduction):
 # 1. lambda x: x
 # 2. grouped-reduce on unique coords is identical to array
 # 3. group_over == groupby-reduce along other dimensions
+
+
+def test_groupby_transpose():
+    # GH5361
+    data = xr.DataArray(
+        np.random.randn(4, 2),
+        dims=["x", "z"],
+        coords={"x": ["a", "b", "a", "c"], "y": ("x", [0, 1, 0, 2])},
+    )
+    first = data.T.groupby("x").sum()
+    second = data.groupby("x").sum()
+
+    assert_identical(first, second.transpose(*first.dims))