What are examples of complexity classes that have contradictory relativizations but they were proven to be either equal or unequal?
$\mathsf{MA_{EXP}} \not\subseteq \mathsf{P/poly}$ but there is an oracle relative to which this is false; both were proved in 


  H. Buhrman, L. Fortnow, T. Thierauf. Nonrelativizing separations. CCC '98. (freely available author's version)