As is well known, the conjugacy classes of the free group $F_2$ are parametrised by cyclically reduced words, up to cyclic permutation. In particular, it's easy to tell whether two elements of $F_2$ are conjugate.

What about the *automorphism* classes of $F_2$? For $u,v\in F_2$ write $u\sim v$ if there is an automorphism of $F_2$ mapping $u$ to $v$. Is there a similarly simple description of a representative from each $\sim$ class?

For example, if $F_2 = \langle x,y\rangle$ then $xyxyxy\sim xxx$ while $xyxyxyx\sim x$.

In particular, is it easy to tell whether $u\sim v$ for general words $u$ and $v$?