  We work out the junction conditions for the Palatini $f(\mathcal{R},T)$
extension of General Relativity, where $f$ is an arbitrary function of the
curvature scalar $\mathcal{R}$ of an independent connection, and of the trace
$T$ of the stress-energy tensor of the matter fields. We find such conditions
on the allowed discontinuities of several geometrical and matter quantities,
some of which depart from their metric counterparts, and in turn extend their
Palatini $f(\mathcal{R})$ versions via some new $T$-dependent terms. Moreover,
we also identify some "exceptional cases" of $f(\mathcal{R},T)$ Lagrangians
such that some of these conditions can be discarded, thus allowing for further
discontinuities in $\mathcal{R}$ and $T$ and, in contrast with other theories
of gravity, they are shown to not give rise to extra components in the matter
sector e.g. momentum fluxes and double gravitational layers. We discuss how
these junction conditions, together with the non-conservation of the
stress-energy tensor ascribed to these theories, may induce non-trivial changes
in the shape of specific applications such as traversable thin-shell wormholes.
