We analytically determine two of the mechanisms which cause chaotic dynamics to appear in a model of the forced and damped Sine-Gordon equation. In particular, we find orbits homoclinic to periodic orbits, and orbits homoclinic to fixed points which satisfy conditions sufficient to guarantee the existence of nearby chaotic invariant sets. One of these homoclinic orbits is a so-called Silnikov-type loop. A proof the existence of a symmetric pair of such loops is our main result. This proof consists of a modified Melnikov perturbation analysis, augmented by some techniques from the field of geometric singular perturbation theory.