  We exhaustively analyze the toric symmetries of CP^3 and its toric blowups.
Our motivation is to study toric symmetry as a computational technique in
Gromov-Witten theory and Donaldson-Thomas theory. We identify all nontrivial
toric symmetries. The induced nontrivial isomorphisms lift and provide new
symmetries at the level of Gromov-Witten Theory and Donaldson-Thomas Theory.
The polytopes of the toric varieties in question include the permutohedron, the
cyclohedron, the associahedron, and in fact all graph associahedra, among
others.
