Ab initio calculations of five two-dimensional intermolecular potential energy surfaces of the Ne-HCN dimer have been performed using the symmetry-adapted perturbation theory and the supermolecular method at different levels of electron correlation. A basis set of spdf-symmetry orbitals (including midbond functions) was used. HCN was assumed linear with interatomic distances fixed at their vibrationally averaged (-1/2) values. Fits to all calculated potential energy surfaces were obtained in the form of angular expansions incorporating the ab initio asymptotic coefficients. It has been found that high-order correlation effects are very important for Ne-HCN and contribute about 20% to the well depth. All of the five surfaces feature a global minimum at the linear Ne-HCN geometry and a narrow and relatively flat valley surrounding HCN. Rovibrational calculations on the surfaces yielded rotational spectra and a rotational constant whose relative differences from their experimental counterparts range from 2% to 12% depending on the method used to obtain the surface. This large sensitivity of spectral quantities to relatively modest differences between the potentials is related to the unusual shape of the potential well.