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We present a general analysis of the bifurcation sequences of periodic orbitsin general position of a family of reversible 1:1 resonant Hamiltonian normalforms invariant under $\Z_2\times\Z_2$ symmetry. The rich structure of theseclassical systems is investigated both with a singularity theory approach andgeometric methods. The geometric approach readily allows to find anenergy-momentum map describing the phase space structure of each member of thefamily and a catastrophe map that captures its global features. Quadratureformulas for the actions, periods and rotation number are also provided.

An energy-momentum map for the time-reversal symmetric 1:1 resonance with Z_2 X Z_2 symmetry