This paper presents the design of a compact gantry that uses superconducting bending magnets (BMs), for use in carbon-ion beam therapy. The size of the gantry is comparable to those of existing gantries that are used for proton-beam therapy. The designed gantry provides point-to-parallel scanning over an area of 20 cm×20 cm at the isocenter, and has rotationally invariant optics, which are enabled by quadrupole and dipole magnets together with a 90° combined-function magnet with 18.6-cm bore radius. A 90° BM accommodates large scanning angles; it also provides equal focusing in horizontal and vertical planes, and zero-integrated nonlinear fields to minimize beam distortion at the isocenter. Three-dimensional field analysis of the magnet, and particle-tracking simulation, validate the beam optics of the gantry and point-to-parallel scanning. The Taylor map and the Lie map are shown to be useful in the analysis of magnetic fields and in optimizing the coil windings.