  We characterize geometrically the semigroups generated by non-selfadjoint
quadratic differential operators $(e^{-tq^w})_{t\geq 0}$ enjoying local
smoothing effects and providing gains of integrability. More precisely, we
prove that the evolution operators $e^{-tq^w}$ map $L^{\mathfrak{p}}$ on
$L^{\mathfrak{q}} \cap C^\infty$, for all $1\leq \mathfrak{p} \leq \mathfrak{q}
\leq \infty$, if and only if the singular space of the quadratic operator $q^w$
is included in the graph of a linear map. We also provide quantitative
estimates for the associated operator norms in the short-time asymptotics $0<t
\ll 1$.
