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We consider efficiently routing permutations in a class of switch-based interconnects. Permutation is an important communication pattern in parallel and distributed computing systems. We present a generic approach to realizing arbitrary permutations in a class of unique-path, self-routable interconnects. It is well-known that this type of interconnect has low hardware cost, but can realize only a small portion of all possible permutations between its inputs and outputs in a single pass. We consider routing arbitrary permutations with link-disjoint paths and node-disjoint paths in such interconnects in a minimum number of passes. In particular, routing with node-disjoint paths has important applications in emerging optical interconnects. We employ and further expand the Latin square technique used in all-to-all personalized exchange algorithms for this class of interconnects for general permutation routing. Our implementation of permutation routing is optimal in terms of the number of passes that messages are transmitted through the network, and it is near-optimal in network transmission time for sufficiently long messages. The possibility of adopting a single-stage interconnect is also discussed.