  The topology of electronic states in band insulators with mirror symmetry can
be classified in two different ways. One is in terms of the mirror Chern
number, an integer that counts the number of protected Dirac cones in the
Brillouin zone of high-symmetry surfaces. The other is via a $\mathbb{Z}_2$
index that distinguishes between systems that have a nonzero quantized orbital
magnetoelectric coupling ("axion-odd"), and those that do not ("axion-even");
this classification can also be induced by other symmetries in the magnetic
point group, including time reversal and inversion. A systematic
characterization of the axion $\mathbb{Z}_2$ topology has previously been
obtained by representing the valence states in terms of hybrid Wannier
functions localized along one chosen crystallographic direction, and inspecting
the associated Wannier band structure. Here we focus on mirror symmetry, and
extend that characterization to the mirror Chern number. We choose the
direction orthogonal to the mirror plane as the Wannierization direction, and
show that the mirror Chern number can be determined from the winding numbers of
the touching points between Wannier bands on mirror-invariant planes, and from
the Chern numbers of flat bands pinned to those planes. In this representation,
the relation between the mirror Chern number and the axion $\mathbb{Z}_2$ index
is readily established. The formalism is illustrated by means of $\textit{ab
initio}$ calculations for SnTe in the monolayer and bulk forms, complemented by
tight-binding calculations for a toy model.
