There is a well-known algorithm for decomposing any given permutation as a product of (not necessarily disjoint) $2$-cycles/transpositions. Such a decomposition of a given $f\circ g$ would, in general, give strong hints about (if not completely determine) the nature of $f$ and $g$.
Why?
Because...
Please would the moderators delete it for me? (I've flagged it already but still . . . )
A permutation $f$ is an involution if $f\circ f=id$.
As you know, any permutation can be written as a product of disjoint cycles; your permutation is $(1\ 10)(2\ 4\ 7)(3\ 5\ 8\ 6\ 9)$. In order to write an arbitrary permutation as a product of two involutions, it suffices (since disjoint permuta...
There is a well-known algorithm for decomposing any given permutation as a product of (not necessarily disjoint) $2$-cycles/transpositions. Such a decomposition of a given $f\circ g$ would, in general, give strong hints about (if not completely determine) the nature of $f$ and $g$.
Why?
Because...
