Is there any relationship between the topologies of the clique complex and the independence complex?

Given any graph $G$, let $G'$ denote $G$ with a new vertex $v$ adjacent to every vertex of $G$. The clique complex of $G'$ is contractible, while the independence complex of $G$ and $G'$ are the same except for an isolated vertex. Similarly we can adjoin a new isolated vertex $w$ to $G$, obtaining a graph $G''$ with a contractible independence complex but with the same clique complex as $G$ except for an isolated vertex. Thus in general there is not much connection between the topologies of the clique and independence complexes.

This is not quite an answer to your question, but still.

There is a version of Alexander duality between homology groups of clique complex $X(G)$ and complex of complements of non-cliques $X^*(G)$, see, e.g. http://arxiv.org/pdf/0710.1172.pdf for precise statement in the case of arbitrary simplicial complexes.