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, a 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.