We apply Thurston's characterization of postcritically finite rational maps as branched coverings of the sphere to give new classes of combination theorems for postcritically finite rational maps. Our constructions increase the degree of the map but always yield branched coverings which are equivalent to rational maps, independent of the combinatorics of the original map. The main tool is a general theorem based on the intersection number of arcs and curves which controls the region in the sphere in which an obstruction may reside.