  We establish the existence of a symmetry within the Gromov-Witten theory of
$\mathbb{CP}^n$ 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.
