Motivated by a question in Schubert calculus, we study various aspects of the divided symmetrization operator, which was introduced by Postnikov in the context of volume polynomials of permutahedra. Divided symmetrization is a linear form which acts on the space of polynomials in n indeterminates of degree n−1. Our main results are related to quasisymmetric polynomials: First, we show that divided symmetrization applied to a quasisymmetric polynomial in m < n indetermi-nates has a natural interpretation. Then, that the divided symmetrization of any polynomial can be naturally computed with respect to a direct sum decomposition due to Aval-Bergeron-Bergeron, involving the ideal generated by positive degree quasisymmetric polynomials in n indeterminates. Several examples with a strong combinatorial flavor are given.