  Effective field theory (EFT) provides a model-independent framework for
interpreting the results of dark matter (DM) direct detection experiments. In
this study, we demonstrate that the two fermionic DM-quark tensor operators,
$(\bar{\chi} i\sigma^{\mu\nu} \gamma^5 \chi) (\bar{q} \sigma_{\mu\nu}q)$ and
$(\bar{\chi} \sigma^{\mu\nu} \chi) (\bar{q} \sigma_{\mu\nu} q)$, can contribute
to the DM electric and magnetic dipole moments via nonperturbative QCD effects,
in addition to the well-studied contact DM-nucleon operators. We then
investigate the constraints on these two operators by considering both the
contact and the dipole contributions using the XENON1T nuclear recoil and
Migdal effect data. We also recast other existing bounds on the DM dipole
operators, derived from electron and nuclear recoil measurements in various
direct detection experiments, as constraints on the two tensor operators. For
$m_\chi \lesssim 1\,\rm GeV$, our results significantly extend the reach of
constraints on the DM-quark tensor operators to masses as low as $5\,\rm MeV$,
with the bound exceeding that obtained by the Migdal effect with only contact
interactions by an order of magnitude or so. In particular, for the operator
$(\bar{\chi} \sigma^{\mu\nu}i\gamma_5 \chi) (\bar{q} \sigma_{\mu\nu}q)$ with DM
mass $m_\chi \gtrsim 10\,\rm GeV$, the latest PandaX constraint on the DM
electric dipole moment puts more stringent bounds than the previous direct
detection limit.
