A reversible mechanical system with a parameter is considered. Periodic motions are investigated. It is shown that symmetric periodic motion continues globally with respect to the parameter. For the special case of a Lagrangian system subjected to positional forces, the existence of a global family of symmetric periodic motions, which originates from equilibrium, is established. In a reversible mechanical system with cyclic coordinates (Routh equations), it is shown that in symmetric invariant sets, including families of periodic motions, the directions of motion are mutually opposite. A scenario is presented for the convergence of global families of periodic motions as the cyclic constants vary. In the problem of the motion of a heavy rigid body with a fixed point, it is found that for a parameter value corresponding to the center of gravity in the principal plane of the ellipsoid of inertia, a bifurcation occurs, giving rise to global families of pendulum oscillations. The families are superimposed on permanent rotations and connect stable and unstable rotations. They continue globally with respect to the constant of the kinetic momentum. In the restricted three-body problem, by introducing a parameter, the existence of a global family of symmetric satellite orbits near a planet is proven.