  We prove that the Cayley graphs $X(G,S)$ and $X^+(G,S)$ are equienergetic for
any abelian group $G$ and any symmetric subset $S$. We then focus on the family
of unitary Cayley graphs $G_R=X(R,R^*)$, where $R$ is a finite commutative ring
with identity. We show that under mild conditions, $\{G_R, G_R^+\}$ are pairs
of integral equienergetic non-isospectral graphs (generically connected and
non-bipartite). Then, we obtain conditions such that $\{G_R, \bar G_R\}$ are
equienergetic non-isospectral graphs. Finally, we characterize all integral
equienergetic non-isospectral triples $\{G_R, G_R^+, \bar G_R \}$ such that all
the graphs are also Ramanujan.
