An improved intermolecular potential surface for the benzene dimer is constructed from interaction energies computed by symmetry-adapted perturbation theory, SAPT(DFT), with the inclusion of third-order contributions. Twelve characteristic points on the surface have been investigated also using the coupled-cluster method with single, double, and perturbative triple excitations, CCSD(T), and triple-zeta quality basis sets with midbond functions. The SAPT and CCSD(T) results are in close agreement and provide the best representation of these points to date. The potential was used in calculations of vibration–rotation-tunneling (VRT) levels of the dimer by a method appropriate for large amplitude intermolecular motions and tunneling between multiple equivalent minima in the potential. The resulting VRT levels were analyzed with the use of the permutation-inversion full cluster tunneling (FCT) group G₅₇₆ and a chain of subgroups that starts from the molecular symmetry group C_s(M) of the rigid dimer at its equilibrium C_s geometry and leads to G₅₇₆ if all possible intermolecular tunneling mechanisms are feasible. Further information was extracted from the calculated wave functions. It was found, in agreement with the experimental data, that for all of the 54 G₅₇₆ symmetry species (with different nuclear spin statistical weights) the lower VRT states have a tilted T-shape (TT) structure; states with the parallel-displaced structure are higher in energy than the ground state of A⁺₁ symmetry by at least 30 cm⁻¹. The dissociation energy D₀ equals 870 cm⁻¹, while the depth D_e of the TT minimum in the potential is 975 cm⁻¹. Hindered rotation of the cap in the TT structure and tilt tunneling lead to level splittings on the order of 1 cm⁻¹. Also intermolecular vibrations with excitation energies starting at a few cm⁻¹ were identified. A further small, but probably significant, level splitting was assigned to cap turnover, although in scans of the potential surface we could not find a plausible ‘reaction path’ for this process. Rotational constants were extracted from energy levels calculated for total angular momentum J = 0 and 1, and from expectation values of the inertia tensor. Although the end-over-end rotational constant B + C agrees well with the measured microwave spectra, there is disagreement with the measurements concerning the (a)symmetric rotor character of the benzene dimer. It is concluded from calculations for the 54 nuclear spin species that the microwave spectrum should show overlapping contributions from many different species. Another interesting conclusion regards the role of the quantum number K, for a prolate near-symmetric rotor the projection of the total angular momentum on the prolate axis. For the benzene dimer, K has a substantial effect on the energy levels associated with the intermolecular motions of the complex.