The observation of strongly-correlated states in moiré systems has renewed the conceptual interest in magnetic systems with higher SU(4) spin symmetry, e.g. to describe Mott insulators where the local moments are coupled spin-valley degrees of freedom. In this talk, we will discuss the physics of a number of SU(4) spin-valley Hamiltonians relevant to moiré materials and make connections also to spin-orbit entangled Kugel-Khomskii models that bear a close resemblance and have a long history in the study of transition metal oxides. Technically, our calculations are based on a pseudo-fermion FRG approach, which has been expanded to tackle diagonal and off-diagonal couplings of generic spin-valley models in the self-conjugate representation of the SU(4) algebra.