We show that initial-state interactions contribute to the cos 2phi distribution in unpolarized Drell-Yan lepton pair production pp and p (p) over bar-->l(+)l(-)X, without suppression. The asymmetry is expressed as a product of chiral-odd distributions h(1)(perpendicular to)(x(1),p(perpendicular to)(2))x (h) over bar (perpendicular to)(1)(x(2),k(perpendicular to)(2)), where the quark-transversity function h(1)(perpendicular to)(x,p(perpendicular to)(2)) is the transverse momentum dependent, light-cone momentum distribution of transversely polarized quarks in an unpolarized proton. We compute this (naive) T-odd and chiral-odd distribution function and the resulting cos 2phi asymmetry explicitly in a quark-scalar diquark model for the proton with initial-state gluon interaction. In this model the function h(1)(perpendicular to)(x,p(perpendicular to)(2)) equals the T-odd (chiral-even) Sivers effect function f(1T)(perpendicular to)(x,p(perpendicular to)(2)). This suggests that the single-spin asymmetries in the semi-inclusive deep inelastic scattering and the Drell-Yan process are closely related to the cos 2phi asymmetry of the unpolarized Drell-Yan process, since all can arise from the same underlying mechanism. This provides new insight regarding the role of the quark and gluon orbital angular momentum as well as that of initial- and final-state gluon exchange interactions in hard QCD processes.