  Graphene heterobilayers with layer antisymmetric strains are studied using
the Dirac-Harper model which describes a pair of single layer Dirac
Hamiltonians coupled by a matrix-valued one-dimensional moir\'e-periodic
interlayer potential. We find that this model hosts low energy, nearly
dispersionless bands near charge neutrality that support anomalous
polarizations of multipole distributions of the charge density. These are
studied by formulating symplectic two-forms that encode field-induced multipole
dynamics allowed in a chiral medium with time reversal symmetry. The model
identifies reciprocity relations between the responses to layer-symmetric and
layer-antisymmetric in-plane electric fields and reveals momentum-space quantum
oscillations that can quantify the structure of its laterally modulated band
inversions.
