  Motivated by recent low-temperature magnetoresistance measurements in twisted
bilayer graphene aligned with hexagonal Boron Nitride substrate, we perform a
systematic study of possible symmetry breaking orders in this device at a
filling of two electrons per Moir\'e unit cell. We find that the surprising
non-monotonic dependence of the resistance on an out-of-plane magnetic field is
difficult to reconcile with particle-hole charge carriers from the low-energy
bands in symmetry broken phases. We invoke the non-zero Chern numbers of the
twisted bilayer graphene flat bands to argue that skyrmion textures provide an
alternative for the dominant charge carriers. Via an effective field-theory for
the spin degrees of freedom, we show that the effect of spin Zeeman splitting
on the skyrmion excitations provides a possible explanation for the
non-monotonic magnetoresistance. We suggest several experimental tests,
including the functional dependence of the activation gap on the magnetic
field, for our proposed correlated insulating states at different integer
fillings. We also discuss possible exotic phases and quantum phase transitions
that can arise via skyrmion-pairing on doping such an insulator.
