  Interactions between Dark Matter (DM) and nucleons relevant for direct search
experiments can be organised in a model independent manner using a Galiliean
invariant, non--relativistic effective field theory (NREFT). Here one expands
the interactions in powers of the momentum transfer $\vec{q}$ and DM velocity
$\vec{v}$. This approach generates many operators. The potentially most
important subleading operators are odd under $T$, and can thus only be present
in a theory with $CP$ violating interactions. We consider two such operators,
called $\mathcal{O}_{10}$ and $\mathcal{O}_{11}$ in the literature, in
simplified models with neutral spin$-0$ mediators; the couplings are chosen
such that the coefficient of the leading spin independent (SI) operator, which
survives for $\vec{v} \rightarrow 0$, vanishes at tree level. However, it is
generically induced at the next order in perturbation theory. We perform a
numerical comparison of the number of scattering events between interactions
involving the $T-$odd operators and the corresponding loop induced SI
contributions. We find that for ''maximal'' $CP$ violation the former can
dominate over the latter. However, in two of the three models we consider, an
electric dipole moment of the neutron (nEDM) is induced at two--loop order. We
find that the experimental bound on the nEDM typically leads to undetectably
small rates induced by ${\mathcal O}_{10}$. On the other hand, the model
leading to a nonvanishing coefficient of ${\mathcal O}_{11}$ does not induce an
nEDM.
