  A class of scalar-tensor theories (STT) including a non-metricity that
unifies metric, Palatini and hybrid metric-Palatini gravitational actions with
non-minimal interaction is proposed and investigated from the point of view of
their consistency with generalized conformal transformations. It is shown that
every such theory can be represented on-shell by a purely metric STT possessing
the same solutions for a metric and a scalar field. A set of generalized
invariants is also proposed. This extends the formalism previously introduced
in \cite{kozak2019}. We then apply the formalism to Starobinsky model, write
down the Friedmann equations for three possible cases: metric, Palatini and
hybrid metric-Palatini, and compare some inflationary observables.
