note that each inclusion is proper.) Indeed, defining new couplings $$\begin{gathered}
c_0\equiv g_0+g_3-1,\qquad  c_1\equiv g_1+g_2-1,\qquad c_2\equiv g_1-g_2,\qquad  c_3\equiv g_0-g_3,\end{gathered}$$ the spectrum of $$\begin{gathered}
\hat{H}(c)\equiv H(g(c))\end{gathered}$$ is invariant under arbitrary permutations of $c_0$, $c_1$, $c_2$, $c_3$.

In Section \[section4\] we consider various special $g$-choices. For the sixteen $g$-values yielding , all of the relevant quantities can be determined in great detail. A striking consequence of our spectral invariance results is that there exist quite a few $g$-choices for which $V(g;x)\ne 0$, yet the spectrum of $H(g)$ equals that for one of the cases where $V(g;x)=0$. We also consider the case $s_g=0$, for which ${{\cal I}}(g)$ has rank 1. In particular, using results from I and [@cadiz], we verify that ${{\cal I}}(g)$ connects the ground state of $H(g')$ with the one of $H(g)$. Finally, we add some observations on the case where the $\pi/r$-periodic potential $V(g;x)$ has no poles for real $x$, a well-known setting that is for instance studied in considerable detail in [@arsc].

The paper is concluded with Appendix \[appendix-A\], in which we present the proofs of Lemmas \[lemma2.1\] and \[lemma2.2\].

The eigenvector ONBs of $\boldsymbol{H(g)}$ and $\boldsymbol{H(g')}$ {#section2}
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Unless explicitly mentioned otherwise, we assume from now on that 