We consider a class of reversible, two-degree of freedom Hamiltonian systems possessing homoclinic orbits to a saddle-center: an equilibrium having two non-zero real and two nonzero imaginary eigenvalues. Under mild nondegeneracy conditions, we construct a two-parameter unfolding and show that there is a countable infinity of secondary homoclinic bifurcations in any neighborhood of the original system. We also demonstrate the existence of families of periodic orbits and of shifts on two symbols (horseshoes). The lack of hyperbolicity and the presence of conserved quantities make the analysis somewhat delicate. We discuss specific examples for which the nondegeneracy conditions can be explicitly checked but indicate that this is not always possible. We illustrate our results with numerical work.