[en] A 2N dimensional symplectic mapping which satisfies the twist condition is obtained from a Lagrangian generating function F(q,q'). The q's are assumed to be angle variables. Reversible maps of this form have 2/sup N+1/ invariant symmetry sets topologically equivalent to N dimensional planes. We conjecture that such maps have at least 2/sup N/ symmetric periodic orbits for each frequency ω = (m,n). Furthermore, we conjecture there is an orbit which minimizes the periodic action for each ω, and 2/sup N/ /minus/ 1 other ''minimax'' orbits. There is ''dominant'' symmetry plane on which the minimizing orbit is never observed to occur. As a parameter varies, the symmetric orbits are observed to undergo symmetry breaking bifurcations, creating a pair nonsymmetric orbits. A pair of coupled standard maps provides a four dimensional example. For this case the two dimensional symmetry planes give a cross section of the resonances and a visualization of the Arnold web