We update van den Bergh's (1976, ApJ, 206, 883) parallel-sequence galaxy classification in which S0 galaxies form a sequence S0a -S0b -S0c that parallels the sequence Sa-Sb-Sc of spiral galaxies. The ratio B/T of bulge-to-total light defines the position of a galaxy in this tuning-fork diagram. Our classification makes one major improvement. We extend the above to an S0a-S0b-S0c-Sph sequence in which spheroidal (Sph) galaxies are juxtaposed to irregular galaxies in a similarly extended Sa-Sb-Sc-Im sequence. This provides a natural "home" for spheroidals, which previously were omitted from galaxy classifications or inappropriately combined with ellipticals. The general conclusion of this work is that environmental secular evolution is more important than we have thought in shaping the structure and evolution of galaxies.