  We present a low-energy model describing the reconstruction of the electronic
spectrum in twisted bilayers of honeycomb crystals with broken sublattice
symmetry. The resulting moir\'e patterns are classified into two families with
different symmetry. In both cases, flat bands appear at relatively large
angles, without any magic angle condition. Transitions between them give rise
to sharp resonances in the optical absorption spectrum at frequencies well
below the gap of the monolayer. Owing to their chiral symmetry, twisted
bilayers display circular dichroism, i.e., different absorption of left and
right circularly-polarized light. This optical activity is a nonlocal property
determined by the stacking. In hexagonal boron nitride, sensitivity to the
stacking leads to strikingly different circular dichroism in the two types of
moir\'es. Our calculations exemplify how subtle properties of the electronic
wavefunctions encoded in current correlations between the layers control
physical observables of moir\'e materials.
