  We formulate a Landau theory for altermagnets, a class of colinear
compensated magnets with spin-split bands. Starting from the non-relativistic
limit, this Landau theory goes beyond a conventional analysis by including
spin-space symmetries, providing a simple framework for understanding the key
features of this family of materials. We find a set of multipolar secondary
order parameters connecting existing ideas about the spin symmetries of these
systems, their order parameters and the effect of non-zero spin-orbit coupling.
We account for several features of canonical altermagnets such as RuO$_2$, MnTe
and CuF$_2$ that go beyond symmetry alone, relating the order parameter to key
observables such as magnetization, anomalous Hall conductivity and
magneto-elastic and magneto-optical probes. Finally, we comment on
generalizations of our framework to a wider family of exotic magnetic systems
deriving from the zero spin-orbit coupled limit.
