  We present a general analysis of the bifurcation sequences of periodic orbits
in general position of a family of reversible 1:1 resonant Hamiltonian normal
forms invariant under $\Z_2\times\Z_2$ symmetry. The rich structure of these
classical systems is investigated both with a singularity theory approach and
geometric methods. The geometric approach readily allows to find an
energy-momentum map describing the phase space structure of each member of the
family and a catastrophe map that captures its global features. Quadrature
formulas for the actions, periods and rotation number are also provided.
