The global relationship between the stellar populations and structural properties of hot galaxies is studied using the same sample of objects analyzed in Bender, Burstein, and Faber. Two measures of global stellar population are used: the Mg2 index at the center of a galaxy and (B-V)0 color measured over a much larger volume. The sample of galaxies studied includes luminous ellipticals that span a wide range in luminosity, compact ellipticals, dwarf ellipticals, and the bulges of S0 galaxies. The degree of anisotropy of internal stellar motions is known for most of these objects. The total sample spans a range of over four orders of magnitude in mass and surface brightness, and the various subclasses populate the fundamental plane in very different ways. Despite this, all galaxies follow the same mean relationship between Mg2 and velocity dispersion sigma0. The relative tightness of this relation is impressive: galaxies show a mean scatter of only 0.025 mag in Mg2, versus a full range of nearly 0.35 mag in Mg2. At least some of the scatter is intrinsic, but the residuals do not correlate with any of the structural properties studied (e.g., velocity anisotropy; effective radius, surface brightness or mass). The residuals also do not show any relation with the positions of the objects within or perpendicular to the fundamental plane. The only properties that do correlate are the morphological and kinematical peculiarities of a handful of disturbed ellipticals, as shown both in this paper and in the study of Schweizer et al. The observed scatter sets an upper limit of 15% on the rms variation of both age and metallicity at fixed sigma0 for bright ellipticals. The Mg2-(B-V)0 relation is also examined and found to be tight and consistent for all dynamically hot galaxies. Several S0 galaxies have much bluer global colors compared to central Mg2 than do other hot galaxies, but these exceptions are likely due to contamination of the global color by young disk light. The generally tight relation between central Mg2 and global (B-V)0 means that variations in the internal color and line-strength gradients from galaxy to galaxy must be small. The Mg2-sigma0 relationship can be reformulated as a function of the mass of the galaxy M and the average volume density of baryonic matter rho as defined by the stars. This new relation can be expressed as Mg2 = (M2rho)0.033. Though rho is used here to denote average volume density, this relation might also be interpreted as a correlation between the local stellar population and local volume density within each dynamically hot galaxy. This prediction will be tested against observed color and line strength gradients in the next paper of this series.