  We show that initial-state interactions contribute to the $\cos 2 \phi$
distribution in unpolarized Drell-Yan lepton pair production $p p$ and $ p \bar
p \to \ell^+ \ell^- X$, without suppression. The asymmetry is expressed as a
product of chiral-odd distributions $h_1^\perp(x_1,\bm{p}_\perp^2)\times \bar
h_1^\perp(x_2,\bm{k}_\perp^2) $, where the quark-transversity function
$h_1^\perp(x,\bm{p}_\perp^2)$ is the transverse momentum dependent, light-cone
momentum distribution of transversely polarized quarks in an {\it unpolarized}
proton. We compute this (naive) $T$-odd and chiral-odd distribution function
and the resulting $\cos 2 \phi$ asymmetry explicitly in a quark-scalar diquark
model for the proton with initial-state gluon interaction. In this model the
function $h_1^\perp(x,\bm{p}_\perp^2)$ equals the $T$-odd (chiral-even) Sivers
effect function $f^\perp_{1T}(x,\bm{p}_\perp^2)$. This suggests that the
single-spin asymmetries in the SIDIS and the Drell-Yan process are closely
related to the $\cos 2 \phi$ asymmetry of the unpolarized Drell-Yan process,
since all can arise from the same underlying mechanism. This provides new
insight regarding the role of quark and gluon orbital angular momentum as well
as that of initial- and final-state gluon exchange interactions in hard QCD
processes.
