  The generalized hybrid metric-Palatini gravity is a theory of gravitation
that has an action composed of a Lagrangian $f(R,\cal R)$, where $f$ is a
function of the metric Ricci scalar $R$ and a new Ricci scalar $\cal R$ formed
from a Palatini connection, plus a matter Lagrangian. This theory can be
rewritten by trading the new geometric degrees of freedom of $f(R,\cal R)$ into
two scalar fields, $\varphi$ and $\psi$, yielding an equivalent scalar-tensor
theory. Given a spacetime theory, the next step is to find solutions. To
construct solutions it is often necessary to know the junction conditions
between two regions at a separation hypersurface $\Sigma$, with each region
being an independent solution. The junction conditions for the generalized
hybrid metric-Palatini gravity are found here, in the geometric and in the
scalar-tensor representations, and in addition, for each representation, the
junction conditions for a matching with a thin-shell and for a smooth matching
at $\Sigma$ are worked out. These junction conditions are applied to three
configurations, a star, a quasistar with a black hole, and a wormhole. The star
has a Minkowski interior, a thin shell at the interface with all the energy
conditions being satisfied, and a Schwarzschild exterior with mass $M$, and for
this theory the matching can only be performed at the shell radius given by
$r_\Sigma=\frac{9M}4$, the Buchdahl radius in general relativity. The quasistar
with a black hole has an interior Schwarzschild black hole surrounded by a
thick shell that matches smoothly to a mass $M$ Schwarzschild exterior at the
light ring, and with the energy conditions being satisfied everywhere. The
wormhole has an interior that contains the throat, a thin shell at the
interface, and a Schwarzschild-AdS exterior with mass $M$ and negative
cosmological constant $\Lambda$, with the null energy condition being obeyed.
