( This may be related to NFAs with more than one initial state )
Consider a dfa $M$ with alphabet $\Sigma$ and states $Q$, typically also characterized by a specific initial state $q_0\in Q$. As the above-cited question points out, nfa's that nondeterministically select among several possible initial states can be converted to equivalent (wrt to their accepted language) nfa's with a unique initial state. And I'm asking if the same is true of dfa's, as follows.
Instead of $M$'s input being only a tape/string $s\in\Sigma^\ast$, let its input be $(q_0,s)\in Q\times\Sigma^\ast$, i.e., you give it the initial state $q_0\in Q$ that you want it to start in, as well as a string.
I'd think each possible $q_0\in Q$ defines a different language, call it $L_{q_0}\subseteq\Sigma^\ast$, accepted by $M$. But $Q$ can possibly be partitioned into equivalence classes, where all the states in a class accept the same $L$. But if there's more than one such class, then no single dfa with a unique $q_0$ starting state could represent this $M$.
So, assuming that's more-or-less correct, my additional question is how to characterize the equivalence classes. That is, given $M$'s transition function, $\delta:\Sigma\times Q\to Q$, how would you determine whether $q\equiv q^\prime$ belong to the same class (besides, that is, individually working out the language each accepts)? Is there a more elegant way to do that?