With new data from the Galactic O-Star Spectroscopic Survey, we confirm and expand the ONn category of late-O, nitrogen-enriched (N), rapidly rotating (n) giants. In particular, we have discovered two "clones" (HD 102415 and HD 117490) of one of the most rapidly rotating O stars previously known (HD 191423, "Howarth's Star"). We compare the locations of these objects in the theoretical H-R diagram to those of slowly rotating ON dwarfs and supergiants. All ON giants known to date are rapid rotators, whereas no ON dwarf or supergiant is, but all ON stars are small fractions of their respective spectral-type/luminosity-class/rotational subcategories. The ONn giants, displaying both substantial processed material and high rotation at an intermediate evolutionary stage, may provide significant information about the development of these properties. They may have preserved high initial rotational velocities or may have been spun up by terminal-age main-sequence core contraction; alternatively, and perhaps more likely, they may be products of binary mass transfer. At least some of them are also runaway stars.