  Guided by non-relativistic Effective Field Theory (EFT) we classify the most
general spin-dependent interactions between a fermionic Weakly Interacting
Massive Particle (WIMP) and nuclei, and within this class of models we discuss
the viability of an interpretation of the DAMA modulation result in terms of a
signal from WIMP elastic scatterings using a halo-independent approach. We find
that, although several relativistic EFT's can lead to a spin-dependent cross
section, in some cases with an explicit, non-negligible dependence on the WIMP
incoming velocity, three main scenarios can be singled out in the
non-relativistic limit which approximately encompass them all, and that only
differ by their dependence on the transferred momentum. For two of them
compatibility between DAMA and other constraints is possible for a WIMP mass
below 30 GeV, but only for a WIMP velocity distribution in the halo of our
Galaxy which departs from a Maxwellian. This is achieved by combining a
suppression of the WIMP effective coupling to neutrons (to evade constraints
from xenon and germanium detectors) to an explicit quadratic or quartic
dependence of the cross section on the transferred momentum (that leads to a
relative enhancement of the expected rate off sodium in DAMA compared to that
off fluorine in droplet detectors and bubble chambers). For larger WIMP masses
the same scenarios are excluded by scatterings off iodine in COUPP.
