Cremona symmetry in Gromov-Witten theory

We establish the existence of a symmetry within the Gromov-Witten theory of ${\mathbb{C}\mathbb{P}}^n$ and its blowup along points. The nature of this symmetry is encoded in the Cremona transform and its resolution, which lives on the toric variety of the permutohedron. This symmetry expresses some difficult to compute invariants in terms of others less difficult to compute. We focus on enumerative implications; in particular this technique yields a one line proof of the uniqueness of the rational normal curve. Our method involves a study of the toric geometry of the permutohedron, and degeneration of Gromov-Witten invariants.