In this work we introduce and study a set of new interleavers based on permutation functions with known inverses over a finite field F_q for using in turbo code structures. We use Möbius and Rédei functions in order to find new interleavers. As a byproduct we give an exact formula for finding the inverse of every Rédei function. The cycle structure of Rédei functions are also investigated. Finally, self-inverse versions of permutation functions are used to construct interleavers. These interleavers are their own de-interleavers and are useful for turbo coding and turbo decoding. Experiments carried out for self-inverse interleavers show excellent agreement with our theoretical results.