  We consider a union of two pants decompositions of the same orientable
2-dimensional surface of any genus g. Each pants decomposition corresponds to
some handlebody bounded by this surface, so two pants decompositions correspond
to a Heegaard splitting of a 3-manifold. We introduce a groupoid FT acting on
double pants decompositions. This groupoid is generated by two simple
transformations (called flips and handle twists), each transformation affecting
only one curve of the double pants decomposition. We prove that FT acts
transitively on all double pants decompositions corresponding to Heegaard
splittings of a 3-dimensional sphere. As a corollary, the mapping class group
is contained in FT.
