  Moir\'e systems provide a highly tunable platform for engineering band
structures and exotic correlated phases. Here, we theoretically study a model
for a single layer of graphene subject to a smooth moir\'e electrostatic
potential, induced by an insulating substrate layer. For sufficiently large
moir\'e unit cells, we find that ultra-flat bands coexist with a triangular
network of chiral one-dimensional (1D) channels. These channels mediate an
effective interaction between localized modes with spin-, orbital- and valley
degrees of freedom emerging from the flat bands. The form of the interaction
reflects the chiralilty and 1D nature of the network. We study this interacting
model within an $SU(4)$ mean-field theory, semi-classical Monte-Carlo
simulations, and an $SU(4)$ spin-wave theory, focusing on commensurate order
stabilized by local two-site and chiral three-site interactions. By tuning a
gate voltage, one can trigger a non-coplanar phase characterized by a peculiar
coexistence of three different types of order: ferromagnetic spin order in one
valley, non-coplanar chiral spin order in the other valley, and 120$^\circ$
order in the remaining spin and valley-mixed degrees of freedom. Quantum and
classical fluctuations have qualitatively different effects on the observed
phases and can, for example, create a finite spin-chirality purely via
fluctuation effects.
