We study weighted azimuthal single spin asymmetries in hadron-hadron scattering using the diagrammatic approach at leading order and assuming factorization. The effects of the intrinsic transverse momenta of the partons are taken into account. We show that the way in which T-odd functions, such as the Sivers function, appear in these processes does not merely involve a sign flip when compared with semi-inclusive deep inelastic scattering, such as in the case of the Drell-Yan process. Expressions for the weighted scattering cross sections in terms of distribution and fragmentation functions folded with hard cross sections are obtained by introducing modified hard cross sections, referred to as gluonic-pole cross sections. © 2005 The American Physical Society.