The air-gap field-effect technique enabled realization of the intrinsic (not limited by static disorder) polaronic transport on the surface of rubrene (C42H28) crystals over a wide temperature range. The signatures of this intrinsic transport are the anisotropy of the carrier mobility, mu, and the growth of mu with cooling. Anisotropy of mu vanishes in the activation regime at low temperatures, where the transport is dominated by shallow traps. The deep traps, introduced by x-ray radiation, increase the field-effect threshold without affecting mu, an indication that the filled traps do not scatter polarons.