  Inelastic dark matter reconciles the DAMA anomaly with other null direct
detection experiments and points to a non-minimal structure in the dark matter
sector. In addition to the dominant inelastic interaction, dark matter
scattering may have a subdominant elastic component. If these elastic
interactions are suppressed at low momentum transfer, they will have similar
nuclear recoil spectra to inelastic scattering events. While upcoming direct
detection experiments will see strong signals from such models, they may not be
able to unambiguously determine the presence of the subdominant elastic
scattering from the recoil spectra alone. We show that directional detection
experiments can separate elastic and inelastic scattering events and discover
the underlying dynamics of dark matter models.
