  We study the prospects for detecting Weakly Interacting Massive Particles
(WIMPs), in a number of phenomenological scenarios, with a detector composed of
a target simultaneously sensitive to both spin-dependent and spin-independent
couplings, as is the case of COUPP (Chicagoland Observatory for Underground
Particle Physics). First, we show that sensitivity to both couplings optimizes
chances of initial WIMP detection. Second, we demonstrate that in case of
detection, comparison of the signal on two complementary targets, such as in
COUPP CF3I and C4F10 bubble chambers, allows a significantly more precise
determination of the dark matter axial and scalar couplings. This strategy
would provide crucial information on the nature of the WIMPs, and possibly
allow discrimination between neutralino and Kaluza-Klein dark matter.
