  Let $m, n$ be positive integers such that $m>1$ divides $n$. In this paper,
we introduce a special class of piecewise-affine permutations of the finite set
$[1, n]:=\{1, \ldots, n\}$ with the property that the reduction $\pmod m$ of
$m$ consecutive elements in any of its cycles is, up to a cyclic shift, a fixed
permutation of $[1, m]$. Our main result provides the cycle decomposition of
such permutations. We further show that such permutations give rise to
permutations of finite fields. In particular, we explicitly obtain classes of
permutation polynomials of finite fields whose cycle decomposition and its
inverse are explicitly given.
