  In this paper we investigate the physical spectrum of the gravitational
theory based on the Poincar\'e group with terms which are at most quadratic in
tetrad and spin connection, allowing for the presence of parity-even as well as
parity-odd invariants. We determine restrictions on the parameters of the
action so that all degrees of freedom propagate and are neither ghosts nor
tachyons. We show that the addition of parity non-conserving invariants extends
the healthy parameter space of the theory. To accomplish our goal, we apply the
weak field approximation around flat spacetime and in order to facilitate the
analysis, we separate the bilinear action for the excitations into completely
independent spin sectors. For this purpose, we employ the spin-projection
operator formalism and extend the original basis built previously, to be able
to handle the parity-odd pieces.
