[en] 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.