diff --git a/xarray/testing/assertions.py b/xarray/testing/assertions.py
index 6418eb79b8b..0a89da3f70e 100644
--- a/xarray/testing/assertions.py
+++ b/xarray/testing/assertions.py
@@ -268,6 +268,10 @@ def _assert_indexes_invariants_checks(
     }
     assert indexes.keys() <= index_vars, (set(indexes), index_vars)
 
+    for k, v in possible_coord_variables.items():
+        if isinstance(v, IndexVariable):
+            assert k == v.name, (k, v.name)
+
     # check pandas index wrappers vs. coordinate data adapters
     for k, index in indexes.items():
         if isinstance(index, PandasIndex):