We prove that a twist-3 soft-gluon-pole (SGP) cross section for single spin asymmetries is determined by a certain "primordial" twist-2 cross section up to kinematic and color factors in the leading order perturbative QCD. In particular, for the processes in which the partonic hard scattering occurs among massless partons, the invariance of the primordial partonic cross section under scale transformation leads to remarkable simplification of the SGP cross section, reproducing the compact form that was recently observed for pion production p(up arrow)p ->pi X and direct-photon production p(up arrow)p ->gamma X.