We investigate theoretically and experimentally the structure of plasmonic modes in individual asymmetric dimer antennas. Plasmonic near-field coupling of high-order modes results in hybridization of bright and dark modes of the individual nanorods, leading to an anticrossing of the coupled resonances. For two bright modes, hybridization results in a capacitive red shift and super-radiant broadening. We show that the properties of asymmetric dimers can be used for nonlinear control of spectral modes and demonstrate such a nonlinear effect by measuring the modulation of a hybrid asymmetric dimer--ITO antenna. With use of full electrodynamical calculations, we find that the properties of the near-field nonlinear responses are distinctly different from the far-field, which opens up new routes for nonlinear control of plasmonic nanosystems.