  We introduce a notion of {\em cyclic Schur-positivity} for sets of
permutations, which naturally extends the classical notion of Schur-positivity,
and it involves the existence of a bijection from permutations to standard
Young tableaux that preserves the cyclic descent set. Cyclic Schur-positive
sets of permutations are always Schur-positive, but the converse does not hold,
as exemplified by inverse descent classes, Knuth classes and conjugacy classes.
  In this paper we show that certain classes of permutations invariant under
either horizontal or vertical rotation are cyclic Schur-positive. The proof
unveils a new equidistribution phenomenon of descent sets on permutations,
provides affirmative solutions to conjectures from [9] and [2], and yields new
examples of Schur-positive sets.

Combinatorics