Multireference configuration interaction (MRD-CI) calculations are reported for a large series of electronic states of the HeNe quasimolecule up to 170000 cm^-1 excitation energy, including those that dissociate to the ³S₁ and 2 ¹S₀ excited states of the He atom. Spin-orbit coupling is included through the use of relativistic effective core potentials (RECPs). Good agreement is obtained with experimental spectroscopic data for the respective atomic levels, although there is a tendency to systematically underestimate the energies of the Ne atom by 1000-1500 cm^-1 because of differences in the correlation effects associated with its ground and Rydberg excited states. Potential curves are calculated for each of these states, and a number of relatively deep minima are found. The CI Ω-state wave functions are sufficiently diabatic until r = 4-5 a₀ to allow for a clear identification of the He ls-2s excited states. Electric dipole transition moments are computed between these states and the HeNe X 0⁺ ground state up to r = 4.0 a₀, and it is found that the 2 ₁S₀ - X maximum value is over an order of magnitude larger than that for the corresponding ³S₁ - X excitation process.