Nondegeneracy of an asymptotic operator under shift and reversal of the loop

Let $S_\infty$ be a continuous $1$-periodic loop of symmetric real $2n\times2n$ matrices such that the asymptotic operator $-J_0\partial_t-S_\infty$ is nondegenerate. Then for $\varepsilon=\pm1$ and every $c\in\mathbb{R}$ the loop $t\mapsto\varepsilon S_\infty(\varepsilon t+c)$ is again a continuous $1$-periodic loop of symmetric matrices, and its asymptotic operator $-J_0\partial_t-\varepsilon S_\infty(\varepsilon t+c)$ is nondegenerate.

For $\varepsilon=1$ this is the invariance of nondegeneracy under a shift of the time parameter; for $\varepsilon=-1$ the loop is also traversed backwards, which is how the asymptotic operator of a negative puncture appears.

Formalization Note Mat n and Vec n are $2n\times2n$ real matrices and $\mathbb{R}^{2n}$, crOp S ζ is $\partial_s\zeta+J_0\partial_t\zeta+S\zeta$, a loop of symmetric matrices is IsSymLoop (continuous, $1$-periodic, symmetric), and nondegeneracy of $-J_0\partial_t-S_\infty$ is CauchyRiemannIndex.IsNondegenerate ($1$ is not an eigenvalue of the time-$1$ map $\Psi(1)$ of $\Psi'=J_0S_\infty\Psi$, $\Psi(0)=1$).