We measure both the linear and nonlinear coupling between transverse modes in a nanomechanical resonator. The nonlinear coupling is due to the displacement dependent tension of the resonator and leads to a frequency shift ("pulling") of each mode proportional to the square of the orthogonal mode s displacement amplitude. The linear coupling is apparent as an avoided crossing of the resonant frequencies that occurs when one electrostatically tunes the modes into degeneracy via a nearby DC gate. We consider the possibility that the linear coupling results from an electrostatic interaction and find that this effect can only partially explain the magnitude of the observed coupling. By measuring the coupled amplitudes magnetomotively at various angles to the applied field, we find that as the modes are tuned through the degeneracy point, they remain linearly polarized, while their planes of vibration rotate by 90 degrees. (C) 2013 AIP Publishing LLC.