Results on the languages recognized by undirected DFAs
I can only give a partial answer. Let $\mathcal{S}$ be the class of all regular languages recognized by a complete SDFA. Then $\mathcal{S}$ is a subclass of the class $\mathcal{G}$ of group languages. A group language is a language  whose syntactic monoid is a finite group , or equivalently, recognized by a finite permutation automaton (each letter induces a permutation on the set of states). The inclusion $\mathcal{S} \subset \mathcal{G}$ is strict. However, the following result holds, showing that $\mathcal{S}$ is a kind of generator for $\mathcal{G}$:

For every group language $L \subseteq A^*$, there is an alphabet $B$, a monoid morphism $f: A^* \to B^*$ and a language $K \subseteq B^*$ in $\mathcal{S}$ such that $L = f^{-1}(K)$. 

Proof. Let $\mathcal{A} = (\{1, ..., n\}, A, \cdot, 1, F)$ be a permutation automaton recognizing $L$. It is a well-known fact that the group of all permutations on $\{1, ..., n\}$ is generated by the set $B$ of its transpositions (a transposition permutes two states and fixes all the other states). By construction, the automaton $\mathcal{B} = (\{1, ..., n\}, B, \cdot, 1, F)$ is a SDFA and hence recognizes a language $K$ of $\mathcal{S}$. Now, for each letter $a \in A$, there is a word $u_a \in B^*$ which defines in $\mathcal{B}$ the same permutation as $a$ in $\mathcal{A}$. Let $f: A^* \to B^*$ be the morphism defined by $f(a) = u_a$ for each letter $a \in A$. It should now be clear that $f^{-1}(K) = L$.

The class $\mathcal{PS}$ of languages recognized by a partial SDFA is of course larger than $\mathcal{S}$, but does not exhaust the class of all regular languages. Actually, one can show that if $L$ is in $\mathcal{PS}$, then the syntactic monoid of $L$ is an inverse monoid and in particular, its idempotents commute. This latter property is shared by the slightly larger class of reversible automata. See

J.-É. Pin, On the languages accepted by finite reversible automata, in 14th ICALP, Berlin, (1987), 237-249, LNCS 267, Springer Verlag