Homogeneous G\"odel-type solutions in hybrid metric-Palatini gravity

[Abridged] If gravitation is to be described by a hybrid metric-Palatini $f(\mathcal{R})$ gravity theory there are a number of issues that ought to be examined in its context, including the question as to whether its equations allow homogeneous G\"odel-type solutions, which necessarily leads to violation of causality. Here, to look further into the potentialities and difficulties of $f(\mathcal{R})$ theories, we examine whether they admit G\"odel-type solutions for well-motivated matter source. We first show that under certain conditions on the matter sources the problem of finding out space-time homogeneous solutions in $f(\mathcal{R})$ theories reduces to the problem of determining solutions of Einstein's field equations with a cosmological constant. Employing this far-reaching result, we determine a general G\"odel-type whose matter source is a combination of a scalar with an electromagnetic field plus a perfect fluid. This general G\"odel-type solution contains special solutions in which the essential parameter $m^2$ can be $m^{2} > 0$, $m=0$, and $m^{2} < 0$, covering thus all classes of homogeneous G\"odel-type spacetimes. This general solution also contains all previously known solution as special cases. The bare existence of these G\"odel-type solutions makes apparent that hybrid metric-Palatini gravity does not remedy causal anomaly in the form of closed timelike curves that are permitted in general relativity.