We analyze a functor from cyclic operads to chain complexes first considered by Getzler and Kapranov and also by Markl. This functor is a generalization of the graph homology considered by Kontsevich, which was defined for the three operads ${\mathrm{Comm}}$, $\mathrm{Assoc}$, and ${\mathrm{Lie}}$. More specifically we show that these chain complexes have a rich algebraic structure in the form of families of operations defined by \emph{fusion} and \emph{fission}. These operations fit together to form uncountably many $\mathrm{Lie}_\infty$ and $\mathrm{co}$-$\mathrm{Lie}_\infty$ structures. In particular, the chain complexes have a bracket and cobracket which are compatible in the {Lie} bialgebra sense on a certain natural subcomplex.