  In many spin glass models, due to the symmetry among sites, any limiting
joint distribution of spins under the annealed Gibbs measure admits the
Aldous-Hoover representation encoded by a function
$\sigma:[0,1]^4\to\{-1,+1\}$, and one can think of this function as a generic
functional order parameter of the model. In a class of diluted models, and in
the Sherrington-Kirkpatrick model, we introduce novel perturbations of the
Hamiltonian that yield certain invariance and self-consistency equations for
this generic functional order parameter and we use these invariance properties
to obtain representations for the free energy in terms of $\sigma$. In the
setting of the Sherrington-Kirkpatrick model, the self-consistency equations
imply that the joint distribution of spins is determined by the joint
distributions of the overlaps, and we give an explicit formula for $\sigma$
under the Parisi ultrametricity hypothesis. In addition, we discuss some
connections with the Ghirlanda-Guerra identities and stochastic stability and
describe the expected Parisi ansatz in the diluted models in terms of $\sigma$.
