Theorem: Partial-product reversal

The declared tensor domain is stable under

and

with real formal $\hslash$.

Proof. Conjugation exchanges the locked first stages and sends ${\mathcal{L}}_2$ to ${\mathcal{R}}_2$, and ${\mathcal{L}}_4$ to ${\mathcal{R}}_4$. Thus it exchanges each left second-stage domain with its right counterpart. Kernel conjugation and $x,x^{\prime }$-exchange send the one-edge and two-edge entries to the corresponding reversed entries. In particular the two mixed metric-quadratic trees are exchanged. The parent dagger symmetry and $c\in \mathbb{R}$ give

The old entries obey the inherited Hermitian convention. The unique glued maps therefore satisfy reversal.