Let $X(G,S)$ be the (undirected) Cayley graph, with $G$ group and $S \subseteq G$ such that $1_G \notin S$ and $S=S^{1}$. Is the complement of $X$ a Cayley graph?

The complete graph on $G$ vertices is a Cayley graph for $S=G\setminus\{1_G\}$. Its complement, the graph without edges, is not a Cayley graph. 

