Spectral symmetry under exchange of couplings

Let $H_N(x,y)$ be the spin-chain Hamiltonian with twist angle $\phi$, depending on couplings $x$ and $y$. The spectral exchange conjecture. For twist angle $\phi=\pi$, the spectrum of $H_N(x,y)$ is invariant under exchanging $x$ and $y$. The statement is described as less obvious than the local tensor-product symmetries and is used later to relate zero-energy states.