It is not necessarily true that Borel equivalence relations that are potentially in the same pointclass are Borel bireducible. For example, consider the orbit equivalence relations of the logic action of $S_{\infty}$ on the standard Borel space of torsion-free abelian groups of rank $n$. Then these orbit equivalence relations are essentially countable and hence are potentially $\mathbf{\Sigma^0_2}$ (for example, see [this][1] theorem). However, Simon Thomas proved that the Borel complexity of isomorphism of torsion-free abelian groups of rank $n$ (strictly) increases with rank (in [this][2] paper).


  [1]: http://mathoverflow.net/questions/215709/natural-examples-of-bf-sigma0-3-equivalence-relations/215743#215743
  [2]: http://www.ams.org/journals/jams/2003-16-01/S0894-0347-02-00409-5/