.

$$\theta_{1-2} = -\theta_{3-4} \equiv \omega$$

(In the original Woodward and Hoffmann book, the example is predicated on the transformation of the sp² terminal hybrids to sp³ hybrids, so that $\sigma$ bonds could be formed between atoms 1 and 4. We choose instead to leave the hybridization alone and allow these terminal carbon atoms to bond using $p_\sigma$ orbitals. Only the symmetry considerations matter, so our approach is good enough. In our simplified approach, we use p_z orbitals to form $p_\sigma$ orbitals). In both cases, the 1-2 and the 3-4 ``$\pi$'' bonds are broken, a 2-3 ``$\pi$'' bond forms, and a 1-4 bond forms ($\sigma$) ( $\theta_{1-2} = \theta_{2-1}$ and $\theta_{3-4} = \theta_{4-3}$ ).