  We prove that 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. This
unveils universal structure behind the SGP cross sections in a variety of hard
processes, and also the special role of the scale invariance in the
corresponding primordial cross section, which leads to remarkable
simplification of the SGP cross sections for the production of massless
particle, such as those for pion production p^\uparrow p \to \pi X and
direct-photon production p^\uparrow p \to \gamma X.
