  We consider within a generalized QCD factorization approach, the high energy
inclusive polarized process p p --> pion + X, including all intrinsic partonic
motions. Several new spin and k_T-dependent soft functions appear and
contribute to cross sections and spin asymmetries. We present here formal
expressions for transverse single spin asymmetries and double longitudinal
ones. The transverse single spin asymmetry, A_N, is considered in detail, and
all contributions are evaluated numerically. It is shown that the azimuthal
phase integrations strongly suppress most contributions, leaving at work mainly
the Sivers effect.
