  The presence of new neutrino-quark interactions can enhance, deplete or
distort the coherent elastic neutrino-nucleus scattering (CEvNS) event rate.
The new interactions may involve CP violating phases that can potentially
affect these features. Assuming light vector mediators, we study the effects of
CP violation on the CEvNS process in the COHERENT sodium-iodine, liquid argon
and germanium detectors. We identify a region in parameter space for which the
event rate always involves a dip and another one for which this is never the
case. We show that the presence of a dip in the event rate spectrum can be used
to constraint CP violating effects, in such a way that the larger the detector
volume the tighter the constraints. Furthermore, it allows the reconstruction
of the effective coupling responsible for the signal with an uncertainty
determined by recoil energy resolution. In the region where no dip is present,
we find that CP violating parameters can mimic the Standard Model CEvNS
prediction or spectra induced by real parameters. We point out that the
interpretation of CEvNS data in terms of a light vector mediator should take
into account possible CP violating effects. Finally, we stress that our results
are qualitatively applicable for CEvNS induced by solar or reactor neutrinos.
Thus, the CP violating effects discussed here and their consequences should be
taken into account as well in the analysis of data from multi-ton dark matter
detectors or experiments such as CONUS, $\nu$-cleus or CONNIE.
