  We consider a general formalism, previously developed to describe single spin
asymmetries (SSA) in inclusive particle production at moderately large $p_T$ in
hadron-hadron high-energy processes. In this approach SSA can originate from
non perturbative, universal, quark distribution analyzing powers, provided the
quark intrinsic motion is taken into account. A fit of the available data on
$p^\uparrow p \to \pi X$ fixes simplified phenomenological expressions of these
quark analyzing powers for the dominant (at large $x_F$) $u$ and $d$ quark
contributions. We use the results of the fit as input to provide a possible
scenario for SSA in a number of interesting processes (e.g. $p p \to \gamma X$,
$p p \to K X$) for which experimental results are or will soon be available.
