A central focus of switching theory is on non-blocking properties of switches and their scalability. Several types of conditionally non-blocking switches have been known to be preserved by the two-stage interconnection network. The preservation of each type is by a separate theory. This study presents a coherent theory that yields an infinite family of preserved types, incorporating all known ones. Recursive two-stage interconnection networks, including all banyan-type networks, construct conditionally non-blocking switches that are most compact in connecting every input to every output and rich in applications.