  Three symmetries prevent a twisted bilayer of graphene from developing an
in-plane spontaneous magnetization in the absence of a magnetic field - time
reversal symmetry, $C_2$ symmetry to $\pi$ rotation and $C_3$ symmetry to
$2\pi/3$ rotation. In contrast, there are experimental and theoretical
indications that, at certain electron densities, time reversal and $C_2$
symmetries are broken spontaneously, while the $C_3$ symmetry is often broken
due to strain. We show that in-plane orbital magnetization is a very sensitive
probe to the simultaneous breaking of these three symmetries, exhibiting
surprisingly large values (of the order of one Bohr magneton per moir\'{e} unit
cell) for valley polarized states at rather small values of heterostrain. We
attribute these large values to the large magnitude of the characteristic
magnetization of individual Bloch states, which we find to reflect the fast
Dirac velocity of single layer graphene, rather than the slow velocity of the
twisted bilayer. These large values are forced to mutually cancel in
valley-symmetric states, but the cancellation does not occur in
valley-polarized states. Our analysis is carried out both for non-interacting
electrons and within a simplified Hartree-Fock framework.

Strongly Correlated Electrons;Mesoscale and Nanoscale Physics