  We study positions of chiral hinge states in higher-order topological
insulators (HOTIs) with inversion symmetry. First, we exhaust all possible
configurations of the hinge states in the HOTIs in all type-I magnetic space
groups with inversion symmetry by studying dependence of the sign of the
surface Dirac mass on surface orientations. In particular, in the presence of
glide symmetry, for particular surface orientations, the surface Dirac mass
changes sign by changing the surface terminations. By applying this result to a
layered antiferromagnet (AFM), we find a difference in the hinge states between
the cases with an even and odd number of layers. In the case of an even number
of layers, which does not preserve inversion symmetry, positions of hinge
states are not inversion symmetric. Nonetheless, these inversion-asymmetric
hinge states result from the bulk topology. We show that their
inversion-asymmetric configurations are uniquely determined from the symmetries
and the topological invariant.
