This follows from the identities
$$A_1^{-1}=UA_1U^{-1},\;\;A_2^{-1}=UA_2U^{-1},$$
$$A_1^{\top}=VA_1V^{-1},\;\;A_2^{\top}=VA_2V^{-1},$$
with
$$U=U^{-1}=\left(
\begin{array}{cccc}
 0 & 1 & 0 & 0 \\
 1 & 0 & 0 & 0 \\
 0 & 0 & 0 & 1 \\
 0 & 0 & 1 & 0 \\
\end{array}
\right),\;\;V=V^{-1}=\left(
\begin{array}{cccc}
 1 & 0 & 0 & 0 \\
 0 & -1 & 0 & 0 \\
 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & -1 \\
\end{array}
\right).$$
Hence if $\lambda\neq 0$ is an eigenvalue of any string of products $M=A_1^{n_1}A_2^{n_2}A_1^{n_3}A_2^{n_4}\cdots A_1^{n_N-1}A_2^{n_N}$ of the two matrices $A_1$ and $A_2$, then also $1/\lambda$ is an eigenvalue:
$${\rm det}\,(\lambda-M)={\rm det}\,(\lambda-M^\top)={\rm det}\,(\lambda-M^{-1}).$$
 The case $\lambda=0$ is excluded because $A_1$ and $A_2$ are nonsingular for any $s$.