In this paper, we propose a new class of interconnection networks, called "biswapped networks" (BSNs). Each BSN is built of 2n copies of some n-node basis network using a simple rule for connectivity that ensures its regularity, modularity, fault tolerance, and algorithmic efficiency. In particular, if the basis network is a Cayley digraph then so is the resulting BSN. Our proposed networks provide a systematic construction strategy for large, scalable, modular, and robust parallel architectures, while maintaining many desirable attributes of the underlying basis network that comprises its clusters. We show how key parameters of a BSN are related to the corresponding parameters of its basis network and obtain a number of results on internode distances, Hamiltonian cycles, and node-disjoint paths. We also discuss the relationship between BSNs and swapped or OTIS networks.