Isomorphism of Borel full groups

Suppose that $G$ and $H$ are Polish groups which act in a Borel fashion on Polish spaces $X$ and $Y$. Let $E_G^X$ and $E_H^Y$ denote the corresponding orbit equivalence relations, and $[G]$ and $[H]$ the corresponding Borel full groups. Modulo the obvious counterexamples, we show that $[G] \cong [H] \Leftrightarrow E_G^X \cong_B E_H^Y$.