\indent We use toric symmetry and degeneration to study the GW theory of $X = {\bf P^1} \times {\bf P^1} \times {\bf P^1}$ . There exists a toric blowup $Y$ of $X$ whose polytope is the permutohedron. The permutohedron admits a symmetry which is manifest in the GW theory of $Y$. We use degeneration to show this symmetry descends to $X$ itself. This also shows a subtle relationship between the GW theories of ${\bf P^1} \times {\bf P^1} \times {\bf P^1}$ and ${\bf P^3}$. All this is joint work with Dhruv Ranganathan.