  We show that Schwarzian theories associated to certain hyperbolic and
parabolic Virasoro coadjoint orbits admit bosonization, i.e. a global
$S^1$-equivariant Darboux chart in which the corresponding path integral
becomes Gaussian. In this chart, correlation functions of bilocals,
time-ordered and out-of-time ordered, can be computed explicitly. We conjecture
that a similar global chart exists for the Teichm\"uller orbit.
