The generating set of Cayley graphs over $Z_n$

Question

Say we have a undirected and connected Cayley graph over $\mathbb{Z}_n$, with generating set $S=\{\pm x_0,\pm x_1,\ldots,\pm x_k\}$. Is it true that we can assume without the loss of generality that $\pm 1 \in S$?

Answer 1

If you can trust my sage calculations, the circulant on 15 vertices with connection set $\{\pm3,\pm5\}$ is not isomorphic to any circulant with connection set $\{\pm1,\pm i\}$ for $i=2,\ldots,7$. (I believe your assumption may be true for circulants of prime power order.)