  We discuss a scenario where the DAMA modulation effect is explained by a
Weakly Interacting Massive Particle (WIMP) which upscatters inelastically to a
heavier state and predominantly couples to the spin of protons. In this
scenario constraints from xenon and germanium targets are evaded dynamically,
due to the suppression of the WIMP coupling to neutrons, while those from
fluorine targets are evaded kinematically, because the minimal WIMP incoming
speed required to trigger upscatters off fluorine exceeds the maximal WIMP
velocity in the Galaxy, or is very close to it. In this scenario WIMP
scatterings off sodium are usually sensitive to the large-speed tail of the
WIMP velocity distribution and modulated fractions of the signal close to unity
arise in a natural way. On the other hand, a halo-independent analysis with
more conservative assumptions about the WIMP velocity distribution allows to
extend the viable parameter space to configurations where large modulated
fractions are not strictly necessary. We discuss large modulated fractions in
the Maxwellian case showing that they imply a departure from the usual cosine
time dependence of the expected signal in DAMA. However we explicitly show that
the DAMA data is not sensitive to this distortion, both in time and frequency
space, even in the extreme case of a 100 % modulated fraction. Moreover the
same scenario provides an explanation of the maximum in the energy spectrum of
the modulation amplitude detected by DAMA in terms of WIMPs whose minimal
incoming speed matches the kinematic threshold for inelastic upscatters. For
the elastic case the detection of such maximum suggests an inversion of the
modulation phase below the present DAMA energy threshold, while this is not
expected for inelastic scattering. This may allow to discriminate between the
two scenarios in a future low-threshold analysis of the DAMA data.
