  The high energy and large p_T inclusive polarized process, (A, S_A) + (B,
S_B) --> C + X, is considered under the assumption of a generalized QCD
factorization scheme. For the first time all transverse motions, of partons in
hadrons and of hadrons in fragmenting partons, are explicitly taken into
account; the elementary interactions are computed at leading order with
noncollinear exact kinematics, which introduces many phases in the expressions
of their helicity amplitudes. Several new spin and k_T dependent soft functions
appear and contribute to the cross sections and to spin asymmetries; we put
emphasis on their partonic interpretation, in terms of quark and gluon
polarizations inside polarized hadrons. Connections with other notations and
further information are given in some Appendices. The formal expressions for
single and double spin asymmetries are derived. The transverse single spin
asymmetry A_N, for p(transv. polarized) p --> pion + X processes is considered
in more detail, and all contributions are evaluated numerically by saturating
unknown functions with their upper positivity bounds. It is shown that the
integration of the phases arising from the noncollinear kinematics strongly
suppresses most contributions to the single spin asymmetry, leaving at work
predominantly the Sivers effect and, to a lesser extent, the Collins mechanism.
