BUG: `MultiIndex.sortlevel` not working after dynamic index construction

[attack68]

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Reproducible Example

.

Issue Description

When constructing this MultiIndex directly the sort works.

>>> midx = MultiIndex.from_tuples([("usd", "usd"), ("eur", "eur")], names=["a", "b"])
>>> midx.sortlevel()
(MultiIndex([('eur', 'eur'),
             ('usd', 'usd')],
            names=['a', 'b']),
 array([1, 0]))

When constructing the index indirectly and dynamically through a loop it fails:

>>> df = DataFrame(index=[0,1], columns=MultiIndex.from_tuples([], names=["a", "b"]))
>>> df.loc[:, ("usd", "usd")] = [10, 20]
>>> df.loc[:, ("eur", "eur")] = [15, 30]
>>> df.columns.sortlevel()
(MultiIndex([('usd', 'usd'),
             ('eur', 'eur')],
            names=['a', 'b']),
 array([0, 1]))

Expected Behavior

I have exactly the same concerns as the OP in issue #49016, which has the same issue after set_levels.

This also occurs on pandas 2.0.0

Incidentally, df.columns.sort_values() does sort the above in all cases, but this is also confusing since sort_values is not documented specifically as a MultiIndex method, but rather an Index method, whereas sort_levels is a MultiIndex method with confusing wording, and no See Also pointers to sort_values.

Installed Versions

1.5.3

[rhshadrach]

but this is also confusing since sort_values is not documented specifically as a MultiIndex method, but rather an Index method, whereas sort_levels is a MultiIndex method

I think this is true across the board - subclasses do not have docs for methods of the parent classes, only the methods that are in addition to the parent.

[attack68]

That does indeed seem true, but a pointer is surely helpful, especially when the method does not work as expected, whether buggy or just poorly described.

[JanHomann]

@attack68 Thanks for picking up the issue again.

@rhshadrach Some of the subclassed Index methods are actually described under both Index and MultiIndex. Index.to_frame is described as MultiIndex.to_frame, same with get_loc, get_indexer, get_level_values, drop, copy, append, and maybe others. So it's not a strict rule.

Also, not all Index methods work on MultiIndex. For example the method Index.shift or the method Index.all do not.

MultiIndex.shift 
NotImplementedError: This method is only implemented for DatetimeIndex, PeriodIndex and TimedeltaIndex; Got type MultiIndex

MultiIndex.all
TypeError: cannot perform all with this index type: MultiIndex

[rhshadrach]

Some of the subclassed Index methods are actually described under both Index and MultiIndex.

I think this is only the case when the subclass overrides the parent. My point is that this is a side-effect of autodoc, not a deliberate choice to exclude this one method from the MultiIndex documentation. If there is a way to add all methods for subclasses, I'd be supportive of that assuming the negative repercussions (build time, docs size) are not too great.

Also, not all Index methods work on MultiIndex. For example the method Index.shift or the method Index.all do not.

I don't understand how this is relevant to this issue, so I might be missing something.