Particle-decaying states of the light nuclei 11,12 N and 12 O were studied using the invariant-mass method. The decay energies and intrinsic widths of a number of states were measured, and the momentum correlations of three-body decaying states were considered. A second 2p-decaying 2 + state of 12 O was observed for the first time, and a higher energy 12 O state was observed in the 4p+2α decay channel. This 4p+2α channel also contains contributions from fission-like decay paths, including 6 Beg.s.+ 6 Beg.s.. Analogs to these states in 12 O were found in 12 N in the 2p+ 10 B and 2p+α+ 6 Li channels. The momentum correlations for the prompt 2p decay of 12 Og.s. were found to be nearly identical to those of 16 Neg.s., and the correlations for the new 2 + state were found to be consistent with sequential decay through excited states in 11 N. The momentum correlations for the 2 + 1 state in 12 O provide a new value for the 11 N ground-state energy. The states in 12 N/ 12 O that belong to the A=12 isobaric sextet do not deviate from the quadratic isobaric multiplet mass equation (IMME) form.