  We show that time-reversal symmetry broken, centrosymmetric antiferromagnets
with nonrelativistic spin-splitting are conveniently described in terms of the
ferroic ordering of magnetic octupoles. The magnetic octupoles are the
lowest-order ferroically ordered magnetic quantity in this case, and so are the
natural order parameter for the transition into the magnetically ordered state.
They provide a unified description of the broken time-reversal symmetry and the
non-relativistic spin splitting as well as a platform for manipulating the
latter, and account for other phenomena, such as piezomagnetism, characteristic
of this class of antiferromagnets. Unusually for antiferromagnets, we show that
the magnetic octupoles cause a non-zero magnetic Compton scattering, providing
a route for their direct experimental detection. We illustrate these concepts
using density-functional and model calculations for the prototypical
non-relativistic spin-split antiferromagnet, rutile-structure manganese
difuoride, MnF2.
