Orthogonal double covers of Cayley graphs

Let X and G be graphs, such that G is isomorphic to a subgraph of X. An orthogonal double cover (ODC) of X by G is a collection B D of subgraphs of X, all isomorphic with G, such that (i) every edge of X occurs in exactly two members of B and (ii) two members of G share an edge if and only if x and y are adjacent in X. The main question is: given the pair (X, G), is there an ODC of X by G? An obvious necessary condition is that X is regular. A technique to construct ODCs for Cayley graphs is introduced. It is shown that for all (X, G) where X is a 3-regular Cayley graph on an abelian group there is an ODC, a few well known exceptions apart.