  In this note we consider the question how the set of inversions of a
permutation $\pi \in S_n$ can be partitioned into two subset, such that those
are itself inversion sets of permutations. This is archived by exploiting a
connection to a graph theoretical result. For this we establish a
correspondence between the substitution decomposition of $\pi$ and the modular
decomposition of its inversion graph. We also consider the special case of
multiplicative decompositions.
