  The transverse polarization distribution of quarks $h_1(x)$ is computed in a
confinement model, the chiral chromodielectric model. The flavor structure of
$h_1$, its $Q^2$ evolution and Soffer's inequality are studied. The Drell--Yan
double transverse asymmetry $A_{TT}$ is evaluated and found to be one order of
magnitude smaller than the double longitudinal asymmetry.
