The Generalized Partition Crossover (GPX) constructs new solutions for the Traveling Salesman Problem (TSP) by finding recombining partitions with one entry and one exit in the graph composed by the union of two parent solutions. If there are k recombining partitions in the union graph, 2^k-2 solutions are simultaneously exploited by GPX. Generalized Asymmetric Partition Crossover (GAPX) is introduced; it finds more recombining partitions and can also find partitions for the asymmetric TSP. GAPX does this by locating partitions that cut vertices of degree 4 in the union graph and by finding partitions with multiple entry and exit points, both in O(n) time. GAPX can improve the quality of solutions generated by the Lin-Kernighan-Helsgaun heuristic and improve the state of the art for the asymmetric TSP.