  We study permutations on $n$ elements preserving orientation (parity) of
every subset of size $k$. We describe all groups of these permutations.
Unexpectedly, these groups (except some special cases) are either trivial or
cyclic or dihedral. We produce these results while defining semigroups
generalizing semigroups of order-preserving mappings and semigroups of
orientation-preserving mappings.
