  The implications of the positivity constraint, $|g_1^{\gamma(P^2)}(x,Q^2)|
\leq F_1^{\gamma(P^2)}(x,Q^2)$, on the presently unknown spin--dependent
structure function $g_1^{\gamma(P^2)}(x,Q^2)$ of real and virtual photons are
studied at scales $Q^2\gg P^2$ where longitudinally polarized photons dominate
physically relevant cross sections. In particular it is shown how to implement
the physical constraints of positivity and continuity at $P^2=0$ in NLO
calculations which afford a nontrivial choice of suitable (DIS) factorization
schemes related to $g_1^{\gamma}$ and $F_1^{\gamma}$ and appropriate boundary
conditions for the polarized parton distributions of real and virtual photons.
The predictions of two extreme `maximal' and `minimal' saturation scenarios are
presented and compared with results obtained within the framework of a simple
quark `box' calculation expected to yield reasonable estimates in the not too
small regions of $x$ and $P^2$.
