  We examine rotational transitions of HCl in collisions with H$_2$ by carrying
out quantum mechanical close-coupling and quasi-classical trajectory
calculations on a recently developed globally accurate full-dimensional ab
initio potential energy surface for the H$_3$Cl system. Signatures of rainbow
scattering in rotationally inelastic collisions are found in the state resolved
integral and differential cross sections as functions of the impact parameter
(initial orbital angular momentum) and final rotational quantum number. We show
the coexistence of distinct dynamical regimes for the HCl rotational transition
driven by the short-range repulsive and long-range attractive forces whose
relative importance depends on the collision energy and final rotational state
suggesting that classification of rainbow scattering into rotational and
$l$-type rainbows is effective for H$_2$+HCl collisions. While the
quasi-classical trajectory method satisfactorily predicts the overall behavior
of the rotationally inelastic cross sections, its capability to accurately
describe signatures of rainbow scattering appears to be limited for the present
system.
