  The so called $f(X)$ hybrid metric-Palatini gravity presents a unique viable
generalisation of the $f(R)$ theories within the metric-affine formalism. Here
the cosmology of the $f(X)$ theories is studied using the dynamical system
approach. The method consists of formulating the propagation equation in terms
of suitable (expansion-normalised) variables as an autonomous system. The fixed
points of the system then represent exact cosmological solutions described by
power-law or de Sitter expansion. The formalism is applied to two classes of
$f(X)$ models, revealing both standard cosmological fixed points and new
accelerating solutions that can be attractors in the phase space. In addition,
the fixed point with vanishing expansion rate are considered with special care
in order to characterise the stability of Einstein static spaces and bouncing
solutions.
