Author’s abstract: We study the \(T\)-periodic solutions to the forced pendulum equation \(u'' + cu' + a\sin u = \lambda f(t)\), where \(f\) satisfies \(f(t + T/2) = -f(t)\). We prove that this equation always has at least two geometrically distinct \(T\)-periodic solutions \(u_0\) and \(u_1\). We then investigate the dependence of the set of \(T\)-periodic solutions on the forced strength \(\lambda\). We prove that under some restriction on the parameters \(a, c\), the periodic solutions found before can be smoothly parameterized by \(\lambda\), and that there are some \(\lambda\)-intervals for which \(u_0(\lambda)\), \(u_1(\lambda)\) are only \(T\)-periodic solutions up to geometrical equivalence, but there are other \(\lambda\)-intervals in which additional \(T\)-periodic solutions bifurcate off the branches. We characterize the global structure of the bifurcating branches. Related to this bifurcation phenomenon is the phenomenon of ‘exchange of stability’ – in some \(\lambda\)-intervals \(u_0(\lambda)\) is dynamically stable and \(u_1(\lambda)\) is unstable, while in other \(\lambda\)-intervals the reverse is true, a phenomenon which has important consequences for the dynamics of the forced pendulum, as we show by both theoretical analysis and numerical simulation.