The near-dissociation microwave rovibronic spectra of HeN+ [Carrington , Chem. Phys. Lett. 262, 598 (1996)] are used to obtain coupled potential energy curves for the six electronic states correlating with He+N+ P-3(0), P-3(1), and P-3(2). High-quality ab initio calculations are carried out, using a spin-restricted open-shell coupled-cluster method with an augmented correlation-consistent quintuple-zeta basis set (aug-cc-pV5Z). Fully coupled calculations of bound and quasibound states are performed, including all six electronic states, and suggest two possible assignments of the observed transitions. The potentials are then morphed (scaled) to reproduce the experimental frequencies. One of the two assignments, designated SH1, is preferred because it gives a more satisfactory explanation of the observed hyperfine splittings. The corresponding morphed potential has well depths of 1954 cm(-1) and 192 cm(-1) for the spin-free (3)Sigma(-) and (3)Pi curves, respectively.