  We develop a correspondence between the study of Borel equivalence relations
induced by closed subgroups of $S_\infty$, and the study of symmetric models
and weak choice principles, and apply it to prove a conjecture of
Hjorth-Kechris-Louveau (1998). For example, we show that the equivalence
relation $\cong^\ast_{\omega+1,0}$ is strictly below
$\cong^\ast_{\omega+1,<\omega}$ in Borel reducibility. By results of
Hjorth-Kechris-Louveau, $\cong^\ast_{\omega+1,<\omega}$ provides invariants for
$\Sigma^0_{\omega+1}$ equivalence relations induced by actions of $S_\infty$,
while $\cong^\ast_{\omega+1,0}$ provides invariants for $\Sigma^0_{\omega+1}$
equivalence relations induced by actions of abelian closed subgroups of
$S_\infty$. We further apply these techniques to study the Friedman-Stanley
jumps. For example, we find an equivalence relation $F$, Borel bireducible with
$=^{++}$, so that $F\restriction C$ is not Borel reducible to $=^{+}$ for any
non-meager set $C$. This answers a question of Zapletal, arising from the
results of Kanovei-Sabok-Zapletal (2013). For these proofs we analyze the
symmetric models $M_n$, $n<\omega$, developed by Monro (1973), and extend the
construction past $\omega$, through all countable ordinals. This answers a
question of Karagila (2019).
