  We study the commensuration torques and layer sliding energetics of
alternating twist trilayer graphene (t3G) and twisted bilayer graphene on
hexagonal boron nitride (t2G/BN) that have two superposed moire interfaces.
Lattice relaxations for typical graphene twist angles of $\sim 1^{\circ}$ in
t3G or t2G/BN are found to break the out-of-plane layer mirror symmetry, give
rise to layer rotation energy local minima dips of the order of $\sim 10^{-1}$
meV/atom at double moire alignment angles, and have sliding energy landscape
minima between top-bottom layers of comparable magnitude. Moire superlubricity
is restored for twist angles as small as $\sim 0.03^\circ$ away from alignment
resulting in suppression of sliding energies by several orders of magnitude of
typically $\sim 10^{-4}$ meV/atom, hence indicating the precedence of rotation
over sliding in the double moire commensuration process.
