  We calculate the perturbative $T$-odd contributions to the lepton angular
distribution in the Drell-Yan process. Using collinear factorization, we work
at the first order in QCD perturbation theory where these contributions appear,
${\cal O}(\alpha_s^2)$, and address both $W^\pm$ and $\gamma/Z^0$ boson
exchange. A major focus of our calculation is on the regime where the boson's
transverse momentum $Q_T$ is much smaller than its mass $Q$. We carefully
expand our results up to next-to-next-to-leading power in $Q_T/Q$. Our
calculation provides a benchmark for studies of $T$-odd contributions that
employ transverse-momentum dependent parton distribution functions. In the
neutral-current case we compare our results for the $T$-odd structure functions
to available ATLAS data.
