Let $R$ be a commutative ring with identity. The involutory Cayley graph $\mathcal{G}(R)$ of $R$ is defined as the graph whose vertex set is the set of elements of $R$, where two vertices $a$ and $b$ are adjacent exactly when $(a-b)^2=1$. This paper investigates the properties of involutory Cayley graphs associated with polynomial and power series rings over the ring of integers modulo $n$.