  We provide a first systematic and quantitative discussion of the
phenomenology of the non-relativistic effective Hamiltonian describing the
nuclear scattering process for a Weakly Interacting Massive Particle (WIMP) of
arbitrary spin $j_\chi$. To this aim we obtain constraints from a
representative sample of present direct detection experiments assuming the
WIMP-nucleus scattering process to be driven by each one of the 44 effective
couplings that arise for $j_\chi\le$ 2. We find that a high value of the
multipolarity $s\le 2 j_\chi$ of the coupling, related to the power of the
momentum transfer $q$ appearing in the scattering amplitude, leads to a
suppression of the expected rates and pushes the expected differential spectra
to large recoil energies $E_R$. For $s\le$ 4 the effective scales probed by
direct detection experiments can be suppressed by up to 5 orders of magnitude
compared to the case of a standard spin-independent interaction. For operators
with large $s$ the expected differential spectra can be pushed to recoil
energies in the MeV range, with the largest part of the signal concentrated at
$E_R\gtrsim$ 100 keV and a peculiar structure of peaks and minima arising when
both the nuclear target and the WIMP are heavy. As a consequence the present
bounds on the effective operators can be significantly improved by extending
the recoil energy intervals to higher recoil energies. Our analysis assumes
effective interaction operators that are irreducible under the rotation group.
Such operators drive the interactions of high-multipole dark matter candidates,
i.e. states that possess only the highest multipole allowed by their spin. As a
consequence our analysis represents also the first phenomenological study of
the direct detection of quadrupolar, octupolar, and hexadecapolar dark matter.
