  We investigate the symmetries of so-called generalized extended CMV matrices.
It is well-documented that problems involving reflection symmetries of standard
extended CMV matrices can be subtle. We show how to deal with this in an
elegant fashion by passing to the class of generalized extended CMV matrices
via explicit diagonal unitaries in the spirit of
Cantero-Gr\"unbaum-Moral-Vel\'azquez. As an application of these ideas, we
construct an explicit family of almost-periodic CMV matrices, which we call the
mosaic unitary almost-Mathieu operator, and prove the occurrence of exact
mobility edges. That is, we show the existence of energies that separate
spectral regions with absolutely continuous and pure point spectrum and exactly
calculate them.
