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Two Boolean functions which differ only by permutation and complementation of their input variables belong to the same symmetry class. Methods are described for determining the number of symmetry classes for functions of variables, and for ascertaining whether or not two functions belong to the same class. This classification is achieved via a complete set of invariants, characteristic of the class, and easily computable from any function in it. The invariants also provide information concerning the size and symmetry properties of the class. Analogous techniques apply to other symmetry classifications of Boolean functions, and to more general categories of discrete mappings.