  We present a theoretical study of dolerophanite Cu$_2$OSO$_4$, a layered
kagome-like spin-$\frac{1}{2}$ magnetic insulator that can be described either
as a system of chains coupled through dimers or as a kagome lattice where every
third spin is replaced by a ferromagnetic spin dimer. Building on insights from
ab initio calculations, classical numerical minimizations, and semiclassical
expansions, we arrive at a minimal microscopic description that accounts for
the experimental data reported so far, including the nature of the magnetic
order, the reported spin length, and the observed anisotropy. The latter arises
by a peculiar competition between the antisymmetric (Dzyaloshinskii-Moriya) and
the symmetric part of the exchange anisotropy, which gives rise to a two-step
re-orientation process involving two successive continuous phase transitions.
Our work uncovers mechanisms stabilizing canted ferrimagnetic order in kagome
systems, and highlights strong magnetic anisotropy in the presence if
dissimilar magnetic orbitals on crystallographically nonequivalent Cu sites. We
also show how these anisotropy terms affect the spin-wave spectrum and how they
can be tracked experimentally.
