  In a previous publication [Beni\' c et al., Phys. Rev. D104 (2021) 094027],
we have computed the perturbative QCD contribution to transverse single spin
asymmetries (SSAs) in semi-inclusive Deep Inelastic Scattering (SIDIS)
involving the $g_T(x)$ distribution. In this paper, we first present a more
efficient derivation of the asymmetries which is applicable to both transverse
and longitudinal SSAs, and correct some inconsistencies in our previous
calculation. We then adapt the method to compute transverse SSAs in Drell-Yan
proportional to $g_T(x)$ and its gluonic counterpart, and discuss the crossing
symmetry between the results for SIDIS and Drell-Yan. Finally, we present
numerical results for various asymmetries measurable at the EIC, RHIC, COMPASS
and Fermilab (SpinQuest), including also part of the genuine twist-three
corrections to $g_T(x)$ from a recent global analysis. We find that the
asymmetries can reach percent-level magnitude, if the kinematics predominantly
probes the large-$x$ (valence) region of the polarized proton, but remain at
sub-percent levels otherwise.
