Alternating subgroup of symmetric group--is it considered a generalization of verbal?

Question

The alternating subgroup ($A_n$) of the symmetric group($S_n$), is, I believe, not a verbal subgroup. But it is generated by elements of the form $xy$, where $x$ and $y$ have the same cycle structure. (Of course, that's equivalent to being generated by the $xy$, where $x$ and $y$ are transpositions.) That's a restriction on the elements $x$, $y$ "allowed" of taking the subgroup generated by all $xy$, a verbal subgroup, namely, the whole group $S_n$. So, is there a generalization of verbal subgroup already defined and studied, which $A_n$ would be an example of, where there you don't quantify over the whole group when defining your words? Does that generalization lead to more general theorems, perhaps slightly weaker than those for verbal subgroups?