Adam isomorphism of circulant graphs

Question

Let $C(n; S)$ denote a circulant graph on $n$ vertices (the vertices can be labeled $0,\ldots,n-1$), and connection set $S = \{s_1, \ldots, s_k \}$. Let $1 \leq \mu < n$ be relatively prime to $n$. It is well known that $C(n; S)$ is isomorphic to $C(n; \mu S)$. Here $\mu S = \{ \mu s \mbox{ (mod } n) | s \in S \}$. This type of isomorphism is called an Adam isomorphism. Now, my question is if it is possible to construct such an isomorphism in all cases where $\mu$ is relatively prime to $n$. For example, could it happen that for some $s \in S$, $\mu s \equiv 0 \mbox{ (mod } n)$?. What would we do in that case? Am I missing something in the definition?