  Axion insulators are magnetic topological insulators in which the non-trivial
$\mathbb{Z}_2$ index is protected by inversion symmetry instead of
time-reversal symmetry. The naturally gapped surfaces of axion insulators give
rise to a half-quantized surface anomalous Hall conductivity (AHC), but the
sign of the surface AHC cannot be determined from topological arguments. In
this paper, we consider topological phenomena at the surface of an axion
insulator. To be explicit, we construct a minimal tight-binding model on the
pyrochlore lattice and investigate the all-in-all-out (AIAO) and ferromagnetic
(FM) spin configurations. We also implement a recently proposed approach for
calculating the surface AHC directly, which allows us to explore how the
interplay between surface termination and magnetic ordering determines the sign
of the half-quantized surface AHC. In the case of AIAO ordering, we show that
it is possible to construct a topological state with no protected metallic
states on boundaries of any dimension (surfaces, hinges, or corners), although
chiral hinge modes do occur for many surface configurations. In the FM case,
rotation of the magnetization by an external field offers a promising means of
control of chiral hinge modes, which can also appear on surface steps or where
bulk domain walls emerge at the surface.
