We apply Thurston's characterization of postcritically finite rationalmaps as branched coverings of the sphere to give new classes ofcombination theorems for postcritically finite rational maps. Ourconstructions increase the degree of the map but always yield branchedcoverings which are equivalent to rational maps, independent of thecombinatorics of the original map. The main tool is a general theorembased on the intersection number of arcs and curves which controlsthe region in the sphere in which an obstruction may reside.