  The combination of space inversion and time reversal symmetries result in
doubly-degenerate Bloch states with opposite spin. Many lattices with these
symmetries can be constructed by combining a noncentrosymmetric potential
(lacking this degeneracy) with its inverted copy. Using simple models, we
unravel the evolution of local spin-splitting during this process of inversion
symmetry restoration, in the presence of spin-orbit interaction and sublattice
coupling. Importantly, through an analysis of quantum mechanical commutativity,
we examine the difficulty of identifying states that are simultaneously
spatially segregated and spin polarized. We also explain how experimental
probes (such as angle-resolved photoemission spectroscopy, or ARPES) of `hidden
spin polarization' in layered materials are susceptible to unrelated spin
splitting intrinsically induced by broken inversion symmetry at the surface.
