We consider the problem of determining the phase-space structure of a Hamiltonian describing a ? : ? resonance. With this we mean a Hamiltonian dynamical system close to an equilibrium with almost equal unperturbed positive frequencies and which is invariant with respect to reflection symmetries in both symplectic variables in addition to the time reversion symmetry. We aim at a general un- derstanding of the bifurcation sequences of periodic orbits in general position from/to normal modes, parametrized by an internal parameter (the ‘energy’) and by the physical parameters: the independent coefficients characterizing the nonlinear perturbation and a ‘detuning’ parameter associated with the quadratic unperturbed Hamiltonian. Among low-order resonances (see, e.g., [??]) the symmetric ? : ? resonance plays a prominent role.