  The interpretation of dark matter direct detection experiments is complicated
by the fact that neither the astrophysical distribution of dark matter nor the
properties of its particle physics interactions with nuclei are known in
detail. To address both of these issues in a very general way we develop a new
framework that combines the full formalism of non-relativistic effective
interactions with state-of-the-art halo-independent methods. This approach
makes it possible to analyse direct detection experiments for arbitrary dark
matter interactions and quantify the goodness-of-fit independent of
astrophysical uncertainties. We employ this method in order to demonstrate that
the degeneracy between astrophysical uncertainties and particle physics
unknowns is not complete. Certain models can be distinguished in a
halo-independent way using a single ton-scale experiment based on liquid xenon,
while other models are indistinguishable with a single experiment but can be
separated using combined information from several target elements.
