Let $M$ be an $(S_n,S_m)$-bimodule. In particular, $M$ is left $S_n$-module, and a right $S_m$-module, and as such admits decompositions into its irreducible $S_n$-modules,
$$M \cong \bigoplus_\lambda V_\lambda,$$
and into its irreducible $S_m$-modules,
$$M \cong \bigoplus_\mu V_\mu.$$
Question. How do you go from this description to give the decomposition of $M$ into
$$M \cong \bigoplus_{\lambda, \mu} V_\lambda \otimes V_\mu$$
(Notation. Here $V_\lambda$ is the irreducible representation corresponding to the partition $\lambda$)
Example. I have an $(S_2,S_4)$-bimodule $M$ of dimension 6. I have a concrete description of the left and right actions, and as such am able to, using character theory, give the following decompositions,
$$M \cong V_{[1,1]}^{\oplus 3} \oplus V_{[2]}^{\oplus 3},$$
and
$$M \cong V_{[1,1,1,1]} \oplus V_{[3,1]} \oplus V_{[2,2]}.$$
I now wish to give the decomposition of the form stated above. I can see that, for dimension reasons, each of the two $S_2$-irreps will have to pair with a 3-dimensional part of the decomposition into $S_4$-irreps. Now $V_{[1,1,1,1]}$ is 1-dimensional, $V_{[2,2]}$ is 2-dimensional, and $V_{[3,1]}$ is 3-dimensional. There are thus two choices,
$$M = V_{[2]}\otimes(V_{[1,1,1,1]}\oplus V_{[2,2]}) \oplus V_{[1,1]}\otimes V_{[3,1]}$$
or the other way around! Is there a systematic, or procedural way to answer this question?