  We present the electronic band structures of states with the same symmetry as
the three-sublattice planar antiferromagnetic order of the triangular lattice.
Such states can also be defined on the honeycomb lattice provided the spin
density waves lie on the bonds. We identify cases which are consistent with
observations on twisted bilayer graphene: a correlated insulator with an energy
gap, yielding a single doubly-degenerate Fermi surface upon hole doping. We
also discuss extensions to metallic states which preserve spin rotation
invariance, with fluctuating spin density waves and bulk $\mathbb{Z}_2$
topological order.
