The inverted plane pendulum is re-examined for the cases of horizontal or vertical driving of the pivot by a harmonic force, with principal interest in the computer simulation of the ensuing motions. While the regions of parameter values for which the vertically driven inverted motion is stable are well understood, and numerical simulation itself is quite stable, the situation is not as clear for the case of horizontal driving. We investigate the origins of these differences from the vertically driven case and resolve them. In doing so, some little-known modes of the horizontally driven hanging pendulum are uncovered and some of their properties are discussed.