  The hybrid metric-Palatini theory of gravity (HMPG), proposed in 2012 by T.
Harko et al., is known to successfully describe both local (solar-system) and
cosmological observations. We discuss static, spherically symmetric vacuum
solutions of HMPG with the aid of its scalar-tensor representation. This
scalar-tensor theory coincides with general relativity with a conformally
coupled scalar field (which can be canonical or phantom), therefore the known
solutions of this theory are re-interpreted in terms of HMPG. In particular, in
the case of zero scalar field potential $V (\phi)$, such that both Riemannian
and Palatini Ricci scalars are zero, generic asymptotically flat solutions
either contain naked singularities or describe traversable wormholes, and there
are only special cases of black hole solutions with extremal horizons. There is
also a one-parameter family of solutions with an infinite number of extremal
horizons between static regions. Examples of analytical solutions with nonzero
potentials $V (\phi)$ are also described, among them black hole solutions with
simple horizons which are generic but, for canonical scalars, they require (at
least partly) negative potentials. With phantom scalars there are ``black
universe'' solutions that lead beyond the horizon to an expanding universe
instead of a singularity. Most of the solutions under consideration turn out to
be unstable under scalar monopole perturbations, but some special black hole
solutions are stable.
