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The structural properties of dynamically hot galaxies are analyzed by combining central velocity dispersion, effective surface brightness, and effective radius into a new 3-space (κ), in which the axes are parameters that are physically meaningful. The degree of velocity dispersion anisotropy is also used. Hot galaxies are found to divide into groups in κ-space that closely parallel conventional morphological classifications: luminous ellipticals, compacts, bulges, bright dwarfs, and dwarf spheroidals. Most systems lie close to the fundamental plane defined by luminous Virgo and Coma ellipticals, indicating similar M/L. However, bulges have somewhat low apparent M/L, while dwarfs have slightly high M/L. The extreme M/L excess of some dwarf spheroidals sets them apart from all other dynamically hot galaxies and indicates strong domination by dark matter. A major sequence is defined by luminous ellipticals, bulges, and most compacts, which together constitute a smooth continuum in κ-space. Several properties vary smoothly with mass along this continuum, including bulge-to-disk ratio, radio properties, rotation, degree of velocity anisotropy, and "unrelaxed." or peculiar, kinematics. These trends are consistent with the idea that the final mergers leading to larger galaxies in this group were systematically more stellar (and less gaseous) than those in smaller galaxies (i.e., a "gas/stellar," or GS, continuum). Structural parameters along this continuum are compared to those predicted for cold dark matter (CDM) in κ-space. CDM predictions can be made to fit the data only if dissipation decreases significantly with increasing mass. This conclusion is consistent with a declining role for gas versus stars with increasing mass as implied by the GS continuum. A second major sequence is comprised of dwarf ellipticals and dwarf spheroidals. These systems populate an elongated locus running at right angles to the main elliptical locus. Various evidence suggests that mass loss is a major factor in hot dwarf galaxies, but the dwarf sequence cannot be simply a mass-loss sequence as it has the wrong direction in κ-space. Hot dwarfs must have come from a range of progenitor galaxies that are not visible today as hot galaxies. The existence of a primarily one-dimensional hot dwarf sequence is surprising and may be at least partially an artifact of selection effects. The most massive and the least massive hot galaxies are anisotropic separated by a strip of galaxies of intermediate mass that are isotropic rotators. The origin of anisotropy in giants and dwarfs is probably different, with that of giants likely being due to stellar mergers, and that in dwarfs possibly being due to expansion following mass loss, or a low rate of internal cloud-cloud collisions due to small collapse factors.