Sambit,

It turned out that there was indeed a bug in 
`parallel::distributed::Triangulation::communicate_locally_moved_vertices` 
in that we didn't update vertices across periodic faces correctly.
PR #5612 <https://github.com/dealii/dealii/pull/5612> Since you provided 
the major part of the added test, I added your name to the corresponding 
changelog entry.

Still, the check for orthogonal_equality doesn't need to suceed even if the 
vertices were moved correctly. We are only updating vertices active cells 
while the PeriodicFacePairs
store CellIterators for the coarsest level. Not all of the vertices of 
these coarse cells are necessarily part of any active cell. Hence, their 
location might not be updated.
Of course the topological information stored in PeriodicFacePairs is not 
changed by moving the mesh, so checking that all the vertices (on ghost 
cells) have been moved correctly should be all you need.
This is also what thet test in PR #5612 does.

Best,
Daniel

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