diff --git a/xarray/core/indexes.py b/xarray/core/indexes.py
index e71c4a6f073..c667124beb7 100644
--- a/xarray/core/indexes.py
+++ b/xarray/core/indexes.py
@@ -701,7 +701,7 @@ def create_variables(
             encoding = None
 
         data = PandasIndexingAdapter(self.index, dtype=self.coord_dtype)
-        var = IndexVariable(self.dim, data, attrs=attrs, encoding=encoding)
+        var = IndexVariable(self.dim, data, attrs=attrs, encoding=encoding, name=name)
         return {name: var}
 
     def to_pandas_index(self) -> pd.Index:
@@ -1153,6 +1153,7 @@ def create_variables(
                 attrs=attrs,
                 encoding=encoding,
                 fastpath=True,
+                name=name,
             )
 
         return index_vars
diff --git a/xarray/core/variable.py b/xarray/core/variable.py
index ec284e411fc..b2e97702ecc 100644
--- a/xarray/core/variable.py
+++ b/xarray/core/variable.py
@@ -2590,12 +2590,14 @@ class IndexVariable(Variable):
     unless another name is given.
     """
 
-    __slots__ = ()
+    __slots__ = ("_name",)
 
     # TODO: PandasIndexingAdapter doesn't match the array api:
     _data: PandasIndexingAdapter  # type: ignore[assignment]
 
-    def __init__(self, dims, data, attrs=None, encoding=None, fastpath=False):
+    def __init__(
+        self, dims, data, attrs=None, encoding=None, fastpath=False, name=None
+    ):
         super().__init__(dims, data, attrs, encoding, fastpath)
         if self.ndim != 1:
             raise ValueError(f"{type(self).__name__} objects must be 1-dimensional")
@@ -2604,6 +2606,11 @@ def __init__(self, dims, data, attrs=None, encoding=None, fastpath=False):
         if not isinstance(self._data, PandasIndexingAdapter):
             self._data = PandasIndexingAdapter(self._data)
 
+        if name is None:
+            self._name = self.dims[0]
+        else:
+            self._name = name
+
     def __dask_tokenize__(self) -> object:
         from dask.base import normalize_token
 
@@ -2753,7 +2760,22 @@ def copy(self, deep: bool = True, data: T_DuckArray | ArrayLike | None = None):
         attrs = copy.deepcopy(self._attrs) if deep else copy.copy(self._attrs)
         encoding = copy.deepcopy(self._encoding) if deep else copy.copy(self._encoding)
 
-        return self._replace(data=ndata, attrs=attrs, encoding=encoding)
+        copied = self._replace(data=ndata, attrs=attrs, encoding=encoding)
+
+        return copied
+
+    def _replace(
+        self,
+        dims=_default,
+        data=_default,
+        attrs=_default,
+        encoding=_default,
+    ) -> Self:
+        replaced = super()._replace(
+            dims=dims, data=data, attrs=attrs, encoding=encoding
+        )
+        replaced._name = self._name
+        return replaced
 
     def equals(self, other, equiv=None):
         # if equiv is specified, super up
@@ -2825,7 +2847,7 @@ def get_level_variable(self, level):
 
     @property
     def name(self) -> Hashable:
-        return self.dims[0]
+        return self._name
 
     @name.setter
     def name(self, value) -> NoReturn:
diff --git a/xarray/testing/assertions.py b/xarray/testing/assertions.py
index 6418eb79b8b..86b7bbdb1e9 100644
--- a/xarray/testing/assertions.py
+++ b/xarray/testing/assertions.py
@@ -264,9 +264,15 @@ def _assert_indexes_invariants_checks(
     }
 
     index_vars = {
-        k for k, v in possible_coord_variables.items() if isinstance(v, IndexVariable)
+        k: v
+        for k, v in possible_coord_variables.items()
+        if isinstance(v, IndexVariable)
     }
-    assert indexes.keys() <= index_vars, (set(indexes), index_vars)
+    index_var_names = set(index_vars)
+    assert indexes.keys() <= index_var_names, (set(indexes), index_var_names)
+
+    for k, v in index_vars.items():
+        assert k == v.name, (k, v.name)
 
     # check pandas index wrappers vs. coordinate data adapters
     for k, index in indexes.items():
@@ -283,7 +289,7 @@ def _assert_indexes_invariants_checks(
         if isinstance(index, PandasMultiIndex):
             pd_index = index.index
             for name in index.index.names:
-                assert name in possible_coord_variables, (pd_index, index_vars)
+                assert name in possible_coord_variables, (pd_index, index_var_names)
                 var = possible_coord_variables[name]
                 assert (index.dim,) == var.dims, (pd_index, var)
                 assert index.level_coords_dtype[name] == var.dtype, (