  We perform a systematic study of the electric and magnetic dipole moments of
dark matter (DM) that are induced at the one-loop level when DM experiences
four-fermion interactions with Standard Model (SM) charged fermions. Related to
their loop nature these moments can largely depend on the UV completion at the
origin of the four-fermion operators. We illustrate this property by
considering explicitly two simple ways to generate these operators, from $t$-
or $s$-channel tree-level exchange. Fixing the strength of these interactions
from the DM relic density constraint, we obtain in particular a magnetic moment
that, depending on the interaction considered, lies typically between
$10^{-20}$ to $10^{-23}$ ecm or identically vanishes. These non-vanishing
values induce, via photon exchange, DM-nucleus scattering cross sections that
could be probed by current or near future direct detection experiments.
