  We show that hematite, $\alpha$-Fe$_2$O$_3$, below its Morin transition, has
a ferroic ordering of rank-5 magnetic triakontadipoles on the Fe ions. In the
absence of spin-orbit coupling, these are the lowest-order ferroically aligned
magnetic multipoles, and they give rise to the $g$-wave non-relativistic spin
splitting in hematite. We find that the ferroically ordered magnetic
triakontadipoles result from the simultaneous antiferroic ordering of the
charge hexadecapoles and the magnetic dipoles, providing a route to
manipulating the magnitude and the sign of the magnetic triakontadipoles as
well as the spin splitting. Furthermore, we find that both the ferroic ordering
of the magnetic triakontadipoles and many of the spin-split features persist in
the weak ferromagnetic phase above the Morin transition temperature.
