Custom indexes and coordinate (re)ordering

What is your issue?

(From https://github.com/pydata/xarray/issues/5647#issuecomment-946546464).

The current alignment logic (as refactored in #5692) requires that two compatible indexes (i.e., of the same type) must relate to one or more coordinates with matching names but also in a matching order.

For some multi-coordinate indexes like PandasMultiIndex this makes sense. However, for other multi-coordinate indexes (e.g., staggered grid indexes) the order of the coordinates doesn't matter much.

Possible options:

1. Setting new Xarray indexes may reorder the coordinate variables, possibly via Index.create_variables(), to ensure consistent order
2. Xarray indexes must implement a Index.matching_key abstract property in order to support re-indexing and alignment.
3. Take care of coordinate order (and maybe other things) inside Index.join and Index.equals, e.g., for PandasMultiIndex maybe reorder the levels beforehand.
   • pros: more flexible
   • cons: not great to implicitly reorder levels if it's a costly operation?
4. Find matching indexes using a two-passes approach: (1) group all indexes by dimension name and (2) check compatibility between the indexes listed in each group.

Start by reading the alignment logic refactored in #5692 and the Index.create_variables(), Index.join(), and Index.equals() entry points named in the issue. Compare the four proposed approaches, then establish a decision and tests covering coordinate reordering for compatible multi-coordinate indexes; the issue is complete when alignment handles the intended coordinate-order cases consistently.