An equivalence relation $E$ is Borel-reducible to an equivalence relation $F$ if there is a $\Delta^1_1$ function $f$ such that $xEy$ holds iff $f(x)Ff(y)$ holds. A set $A\subset \omega^{\omega}$ is Wadge reducible to a set $B\subset \omega^{\omega}$ if there is a continuous function $g$ such that $A=g^-1[B]$. How far can the analogy between Borel-redution and Wadge-reduction be pushed? In particular is there a notion of the Borel order just like the Wadge order of a set of reals? Also can the Martin-Monk argument showing that the relation of Wadge reduction is wellfounded be replicated for the relation of Borel-reduction? If yes do projective ordinal ($AD$ in general) have any consequences on being Borel-reducible?

My motivation is that I just started reading about the area of classification problems and Borel-reduction and some concepts and definition look very similar to the Wadge hierarchy (this is kind of obvious actually because both notions are about descriptive set theoretic complexity).