We present and discuss correlations for optical and near-infrared (5500-10030 Å) line intensity measurements at many positions in the Crab Nebula. These correlations suggest the existence of gas produced by a range of nuclear processing, from material in which synthesis ended with the CNO cycle, to some helium burning and nitrogen depletion, to regions containing enriched products of oxygen burning. The latter exhibit a gradual, linear rise of [Ni II] emission with increasing argon enrichment, whereas gas with less nuclear processing shows markedly different [Ni II] emission characteristics, including the highest derived abundances. This suggests two origins for stable, neutron-rich nickel in the nebula: a type of "alpha-rich freeze out" in the more highly processed material and possibly removal of ions from the neutron star in other regions. In addition, the data indicate that anomalously strong observed [C I] emission comes from broad, low-ionization H+ to H0 transition zones. Although the strongest He I emission could also be enhanced in similar low-ionization gas, correlations between relevant line ratios argue against that interpretation, strengthening the case for an exceptionally high helium mass fraction in some locations.