We carry out a joint spatial-kinematical-metallicity analysis of globular clusters (GCs) around the Andromeda Galaxy (M31), using a homogeneous, high-quality spectroscopic data set. In particular, we remove the contaminating young clusters that have plagued many previous analyses. We find that the clusters can be divided into three major metallicity groups based on their radial distributions: (1) an inner metal-rich group ([Fe/H]>-0.4); (2) a group with intermediate metallicity (with median [Fe/H]=-1); and (3) a metal-poor group, with [Fe/H]. The metal-rich group has kinematics and spatial properties like those of the disk of M31, while the two more metal-poor groups show mild prograde rotation overall, with larger dispersions --in contrast to previous claims of stronger rotation. The metal-poor GCs are the least concentrated group; such clusters occur five times less frequently in the central bulge than do clusters of higher metallicity. Despite some well-known differences between the M31 and Milky Way GC systems, our revised analysis points to remarkable similarities in their chemodynamical properties, which could help elucidate the different formation stages of galaxies and their GCs. In particular, the M31 results motivate further exploration of a metal-rich GC formation mode in situ, within high-redshift, clumpy galactic disks.