\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.