  In this paper, the hybrid metric-Palatini gravity is an approach to modified
gravity in which is added to the usual Einstein-Hilbert action a supplementary
term containing a Palatini-type correction of the form $f({\cal R},T)$. Here,
${\cal R}$ is the Palatini curvature scalar, which is constructed from an
independent connection and $T$ is the trace of the energy-momentum tensor. This
theory describes a non-minimal coupling between matter and geometry. The
modified Einstein field equations in this hybrid metric-Palatini approach are
obtained. Then, it is investigated whether this modified theory of gravity and
its field equations allow G\"{o}del-type solutions, which essentially lead to
violation of causality. Considering physically well-motivated matter sources,
causal and non-causal solutions are explored.
