  Recent experimental results indicate that the dark matter sector may have a
non-minimal structure with a spectrum of states and interactions. Inelastic
scattering has received particular attention in light of DAMA's annual
modulation signal. Composite inelastic dark matter (CiDM) provides a dynamical
origin for the mass splittings in inelastic dark matter models. We show that
higher dimensional operators in the CiDM Lagrangian lead to an admixture of
inelastic and elastic scattering in the presence of parity violation. This
scenario is consistent with direct detection experiments, even when parity
violation is nearly maximal. We present an effective field theory description
of such models and discuss the constraints from direct detection experiments.
The CiDM model with parity violation has non-trivial phenomenology because of
the multiple scattering channels that are allowed.
