id,node_id,number,title,user,state,locked,assignee,milestone,comments,created_at,updated_at,closed_at,author_association,active_lock_reason,draft,pull_request,body,reactions,performed_via_github_app,state_reason,repo,type
626625818,MDExOlB1bGxSZXF1ZXN0NDI0NTgzMjYw,4108,Convert Variable to IndexVariable or vice versa when renamed,2448579,closed,0,,,2,2020-05-28T15:59:53Z,2022-10-18T19:15:54Z,2021-09-23T17:51:16Z,MEMBER,,0,pydata/xarray/pulls/4108,"<!-- Feel free to remove check-list items aren't relevant to your change -->

 - [x] Closes #4107, closes #4417
 - [x] Tests added
 - [x] Passes `isort -rc . && black . && mypy . && flake8`
 - [x] Fully documented, including `whats-new.rst` for all changes and `api.rst` for new API
","{""url"": ""https://api.github.com/repos/pydata/xarray/issues/4108/reactions"", ""total_count"": 0, ""+1"": 0, ""-1"": 0, ""laugh"": 0, ""hooray"": 0, ""confused"": 0, ""heart"": 0, ""rocket"": 0, ""eyes"": 0}",,,13221727,pull
626591460,MDU6SXNzdWU2MjY1OTE0NjA=,4107,renaming Variable to a dimension name does not convert to IndexVariable,2448579,closed,0,4160723,,0,2020-05-28T15:11:49Z,2022-09-27T09:33:42Z,2022-09-27T09:33:42Z,MEMBER,,,,"<!-- A short summary of the issue, if appropriate -->

Seen in #4103

#### MCVE Code Sample
<!-- In order for the maintainers to efficiently understand and prioritize issues, we ask you post a ""Minimal, Complete and Verifiable Example"" (MCVE): http://matthewrocklin.com/blog/work/2018/02/28/minimal-bug-reports -->

```python
from xarray.tests import assert_identical

coord_1 = xr.DataArray([1, 2], dims=[""coord_1""], attrs={""attrs"": True})
da = xr.DataArray([1, 0], [coord_1])
obj = da.reset_index(""coord_1"").rename({""coord_1_"": ""coord_1""})
assert_identical(da, obj)
```

#### Expected Output


#### Problem Description
```
AssertionErrorTraceback (most recent call last)
<ipython-input-19-02ef6bd89884> in <module>
----> 1 assert_identical(da, obj)

~/work/python/xarray/xarray/tests/__init__.py in assert_identical(a, b)
    160     xarray.testing.assert_identical(a, b)
    161     xarray.testing._assert_internal_invariants(a)
--> 162     xarray.testing._assert_internal_invariants(b)
    163 
    164 

~/work/python/xarray/xarray/testing.py in _assert_internal_invariants(xarray_obj)
    265         _assert_variable_invariants(xarray_obj)
    266     elif isinstance(xarray_obj, DataArray):
--> 267         _assert_dataarray_invariants(xarray_obj)
    268     elif isinstance(xarray_obj, Dataset):
    269         _assert_dataset_invariants(xarray_obj)

~/work/python/xarray/xarray/testing.py in _assert_dataarray_invariants(da)
    210     assert all(
    211         isinstance(v, IndexVariable) for (k, v) in da._coords.items() if v.dims == (k,)
--> 212     ), {k: type(v) for k, v in da._coords.items()}
    213     for k, v in da._coords.items():
    214         _assert_variable_invariants(v, k)

AssertionError: {'coord_1': <class 'xarray.core.variable.Variable'>}
```

#### Versions

<details><summary>Output of <tt>xr.show_versions()</tt></summary>

<!-- Paste the output here xr.show_versions() here -->


</details>
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