  In a recent paper, we introduced a new discretization scheme for gravity in
2+1 dimensions. Starting from the continuum theory, this new scheme allowed us
to rigorously obtain the discrete phase space of loop gravity, coupled to
particle-like "edge mode" degrees of freedom. In this work, we expand on that
result by considering the most general choice of integration during the
discretization process. We obtain a family of polarizations of the discrete
phase space. In particular, one member of this family corresponds to the usual
loop gravity phase space, while another corresponds to a new polarization, dual
to the usual one in several ways. We study its properties, including the
relevant constraints and the symmetries they generate. Furthermore, we motivate
a relation between the dual polarization and teleparallel gravity.
