  We extend the formalism of dark matter directional detection to arbitrary
one-body dark matter-nucleon interactions. The new theoretical framework
generalizes the one currently used, which is based on 2 types of dark
matter-nucleon interaction only. It includes 14 dark matter-nucleon interaction
operators, 8 isotope-dependent nuclear response functions, and the Radon
transform of the first 2 moments of the dark matter velocity distribution. We
calculate the recoil energy spectra at dark matter directional detectors made
of CF$_4$, CS$_2$ and $^{3}$He for the 14 dark matter-nucleon interactions,
using nuclear response functions recently obtained through numerical nuclear
structure calculations. We highlight the new features of the proposed
theoretical framework, and present our results for a spherical dark matter halo
and for a stream of dark matter particles. This study lays the foundations for
model independent analyses of dark matter directional detection experiments.
