This paper introduces a new recursion principle for inductively defined data modulo α-equivalence of bound names that makes use of Odersky-style local names when recursing over bound names. It is formulated in simply typed λ-calculus extended with names that can be restricted to a lexical scope, tested for equality, explicitly swapped and abstracted. The new recursion principle is motivated by the nominal sets notion of ‘α-structural recursion’, whose use of names and associated freshness side-conditions in recursive definitions formalizes common practice with binders. The new calculus has a simple interpretation in nominal sets equipped with name-restriction operations. It is shown to adequately represent α-structural recursion while avoiding the need to verify freshness side-conditions in definitions and computations. The paper is a revised and expanded version of Pitts (Nominal System T. In Proceedings of the 37th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages, POPL 2010 (Madrid, Spain). ACM Press, pp. 159–170, 2010).