We explore various techniques to compress a permutation π over n integers, taking advantage of ordered subsequences in π, while supporting its application π(i) and the application of its inverse π^−1(i) in small time. Our compression schemes yield several interesting byproducts, in many cases matching, improving or extending the best existing results on applications such as the encoding of a permutation in order to support iterated applications π^k (i) of it, of integer functions, and of inverted lists and suffix arrays.