A class of Recursive Permutations which is Primitive Recursive complete

We focus on total functions in the theory of reversible computational models. We define a class of recursive permutations, dubbed Reversible Primitive Permutations (RPP) which are computable invertible total endo-functions on integers, so a subset of total reversible computations. RPP is generated from five basic functions (identity, sign-change, successor, predecessor, swap), two notions of composition (sequential and parallel), one functional iteration and one functional selection. RPP is closed by inversion and it is expressive enough to encode Cantor pairing and the whole class of Primitive Recursive Functions.