inverse image of the symmetrization map

Question

Let $S_n$ denote the vector space of $n \times n$ matrices and $M_{m \times n}$ denote the vector space of $m \times n$ matrices. Then we have a map $f: M_{m \times n} \times M_{n \times m} \rightarrow S_n \times S_n$ defined by $(X,Y) \rightarrow (X^TX, YY^T)$ and this gives an algebra map of the corresponding algebras $f^*:\mathbb C[S_m \times S_m] \rightarrow \mathbb C[M_{m \times n} \times M_{n \times m}]$. The image of $f^*$ is $\mathbb C[M_{m \times n} \times M_{n \times m}]^{O_n}$, the subalgebra of polynomials invariant under the orthogonal group $O_n$ and infact this is an isomorphism if $n \geq m$. Now $Tr(AB) \in \mathbb C[M_{m \times n} \times M_{n \times m}]^{O_n}$. Then what is the inverse image of $Tr(AB)$ ? Is it trace of a monomial in $A,A^T, B$ and $B^T$ ?