  The discovery of correlated insulating states in moir\'e heterostructures has
renewed the interest in strongly-coupled electron systems where spin and valley
(or layer) degrees of freedom are intertwined. In the strong-coupling limit,
such systems can be effectively described by SU(4) spin-valley models akin to
Kugel-Khomskii models long studied in the context of spin-orbit coupled
materials. However, typical moir\'e heterostructures also exhibit interactions
that break the SU(4) symmetry down to
SU(2)${}_{\mathrm{spin}}\otimes$U(1)${}_{\mathrm{valley}}$. Here we investigate
the impact of such symmetry-breaking couplings on the magnetic phase diagram
for triangular superlattices considering a filling of two electrons (or holes)
per moir\'e unit cell. We explore a broad regime of couplings -- including XXZ
anisotropies, Dzyaloshinskii-Moriya exchange and on-site Hund's couplings --
using semi-classical Monte Carlo simulations. We find a multitude of
classically ordered phases, including (anti-)ferromagnetic, incommensurate, and
stripe order, manifesting in different sectors of the spin-valley model's
parameter space. Zooming in on the regimes where quantum fluctuations are
likely to have an effect, we employ pseudo-fermion functional renormalization
group (pf-FRG) calculations to resolve quantum disordered ground states such as
spin-valley liquids, which we indeed find for certain parameter regimes. As a
concrete example, we discuss the case of trilayer graphene aligned with
hexagonal boron nitride using material-specific parameters.
