  Antiferromagnets are normally thought of as materials with compensated
magnetic sublattices. This adds to their technological advantages but
complicates readout of the antiferromagnetic state. We demonstrate
theoretically the existence of a Dzyaloshinskii-Moriya interaction (DMI) which
is determined by the magnetic symmetry classes of Cr$_2$O$_3$ surfaces with an
in-plane magnetic easy axis. The DMI explains a previously predicted
out-of-plane magnetization at the nominally compensated surfaces of chromia,
leading to a surface-localized canted ferrimagnetism. This is in agreement with
magnetotransport measurements and with density functional theory predictions
which further allow us to quantify the strength of DMI. The temperature
dependence of the transversal resistance for these planes shows distinct
behavior in comparison with that of the Cr$_2$O$_3$ $c$ plane, which we
attribute to the influence of DMI. Our work provides a framework to analyze
surface-driven phenomena in antiferromagnets, and motivates the use of
nominally compensated chromia surfaces for antiferomagnetic spintronics and
magnonics.
