This question is just for fun and this is completely outside my area, so it's likely dumb; apologies in advance.
By a "quotient" I mean the following: suppose you have two complexity classes, $A \subseteq B$. The quotient $B/A$ would consist of the equivalence classes of elements of $B$ under the relation $b \sim b'$ if you can solve $b'$ with a program from $A$ given input from an oracle for $b$, and vice-versa. (I don't know if this concept has a name or is called something else; sorry.)
(To give the obvious example, the $P$ versus $NP$ problem asks whether $NP/P$ is trivial.)
Can anything interesting be said about this notion?