A statistical analysis of the temporal evolution of hard X-ray (HXR) footpoint motions in 27 M- and X-class solar flares observed by the Reuven Ramaty High Energy Solar Spectroscopic Imager is presented. Extreme UV images from TRACE and SOHO/EIT, H alpha images, and magnetograms from SOHO/MDI are used to put the HXR footpoint motions in context of flare ribbons and the magnetic neutral line. Footpoint motions are often found to be complex making a statistical analysis difficult. In an attempt to simplify the analysis, each event was searched for motions predominantly parallel and predominantly perpendicular to the neutral line or flare ribbons. Four kinds of complex motions are described and their relationships to the possible magnetic reconnection processes are discussed. In the soft X-ray (SXR) rise phase, motions along the neutral line or flare ribbons are most common (20 out of 27) and only two events show perpendicular motions (for the remaining five events a simple classification was not possible). However, at later times around the SXR peak, perpendicular motion is more frequently observed (similar to 40%) than motions along the neutral line or ribbons similar to 27%). The preference of HXR kernels appearing at the footpoints of highly-sheared magnetic loops at the start of the SXR rise phase is consistent with the magnetic reconnection theory that the reconnection occurring at sheared magnetic arcade field lines produces most HXR energy release in the impulsive phase of large flares.