The Hubble classification scheme has a well-known drawback for elliptical galaxies: the sequence E0-E6 correlates primarily with inclination and not with fundamental properties. In contrast, later-type galaxies are ordered by fundamental physical parameters. We propose to revise the Hubble sequence so that it orders ellipticals by isophote shape. Specifically, we suggest that the Im-spiral-S0 sequence be connected to disky ellipticals and thence to boxy ellipticals. The sequence is continuous from S0's to disky E's. However, boxy E's may be unrelated to other ellipticals: global and core properties both show signs of a dichotomy between (1) normal- and low-luminosity ellipticals that rotate rapidly, that are nearly isotropic and oblate-spheroidal, that are coreless, and that have disky-distorted isophotes, and (2) giant ellipticals that are essentially nonrotating, that are anisotropic and moderately triaxial, that have cuspy cores, and that are boxy-distorted. In the classification, we use isophote shape as an implicit indicator of velocity anisotropy. Two observations support this interpretation: (1) ellipticals with disky isophote distortions rotate rapidly while boxy ellipticals tend to rotate slowly; (2) disky ellipticals show little minor-axis rotation while boxy ellipticals often have large amounts of minor-axis rotation. An approximate classification by velocity anisotropy extends the Im-spiral-S0 sequence through the ellipticals in a physically reasonable way, and it is diagnostic of formation mechanisms. Care is required to recognize unfundamental complications, such as the building by accretion of disky substructure inside a boxy elliptical. However, we believe that the proposed scheme distills the essential physics that should be embodied in a relevant morphological classification.
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