  We calculate the Sivers function in semi-inclusive deep inelastic scattering
(SIDIS) and in the Drell-Yan process (DY) by employing the quasi-classical
Glauber-Mueller/ McLerran-Venugopalan approximation. Modeling the hadron as a
large "nucleus" with non-zero orbital angular momentum (OAM), we find that its
Sivers function receives two dominant contributions: one contribution is due to
the OAM, while another one is due to the local Sivers function density in the
nucleus. While the latter mechanism, being due to the "lensing" interactions,
dominates at large transverse momentum of the produced hadron in SIDIS or of
the di-lepton pair in DY, the former (OAM) mechanism is leading in saturation
power counting and dominates when the above transverse momenta become of the
order of the saturation scale. We show that the OAM channel allows for a
particularly simple and intuitive interpretation of the celebrated sign flip
between the Sivers functions in SIDIS and DY.
