  Many magnetic point-group symmetries induce a topological classification on
crystalline insulators, dividing them into those that have a nonzero quantized
Chern-Simons magnetoelectric coupling ("axion-odd" or "topological"), and those
that do not ("axion-even" or "trivial"). For time-reversal or inversion
symmetries, the resulting topological state is usually denoted as a "strong
topological insulator" or an "axion insulator" respectively, but many other
symmetries can also protect this "axion $Z_2$" index. Topological states are
often insightfully characterized by choosing one crystallographic direction of
interest, and inspecting the hybrid Wannier (or equivalently, the non-Abelian
Wilson-loop) band structure, considered as a function of the two-dimensional
Brillouin zone in the orthogonal directions. Here, we systematically classify
the axion-quantizing symmetries, and explore the implications of such
symmetries on the Wannier band structure. Conversely, we clarify the conditions
under which the axion$Z_2$ index can be deduced from the Wannier band
structure. In particular, we identify cases in which a counting of Dirac
touchings between Wannier bands, or a calculation of the Chern number of
certain Wannier bands, or both, allows for a direct determination of the axion
$Z_2$ index. We also discuss when such symmetries impose a "flow" on the
Wannier bands, such that they are always glued to higher and lower bands by
degeneracies somewhere in the projected Brillouin zone, and the related
question of when the corresponding surfaces can remain gapped, thus exhibiting
a half-quantized surface anomalous Hall conductivity. Our formal arguments are
confirmed and illustrated in the context of tight-binding models for several
paradigmatic axion-odd symmetries including time reversal, inversion, simple
mirror, and glide mirror symmetries.
