  Moir\'{e} superlattices in twisted bilayer graphene and transition-metal
dichalcogenides have emerged as a powerful tool for engineering novel band
structures and quantum phases of two-dimensional quantum materials. Here we
investigate Moir\'{e} physics emerging from twisting two independent hexagonal
optical lattices of atomic (pseudo-)spin states (instead of bilayers), which
exhibits remarkably different physics from twisted bilayer graphene. We employ
a momentum-space tight-binding calculation that includes all range real-space
tunnelings, and show that all twist angles $\theta \lesssim 6^{\circ }$ can
become magic that support gapped flat bands. Due to greatly enhanced density of
states near the flat bands, the system can be driven to superfluid by weak
attractive interaction. Strikingly, the superfluid phase corresponds to a
Larkin-Ovchinnikov state with finite momentum pairing, resulting from the
interplay between flat bands and inter-spin interactions in the unique
single-layer spin-twisted lattice. Our work may pave the way for exploring
novel quantum phases and twistronics in cold atomic systems.
