The optical response of a plasmonic gap-antenna is mainly determined by the Coulomb interaction of the two constituent arms of the antenna. Using rigorous calculations supported by simple analytical models, we observe how the morphology of a nanometric gap separating two metallic rods dramatically modifies the plasmonic response. In the case of rounded terminations at the gap, a conventional set of bonding modes is found that red-shifts strongly with decreasing separation. However, in the case of flat surfaces, a distinctly different situation is found with the appearance of two sets of modes: (i) strongly radiating longitudinal antenna plasmons (LAPs), which exhibit a red-shift that saturates for very narrow gaps, and (ii) transverse cavity plasmons (TCPs) confined to the gap, which are weakly radiative and strongly dependent on the separation distance between the two arms. The two sets of modes can be independently tuned, providing detailed control of both the near- and far-field response of the antenna. We illustrate these properties also with an application to larger infrared gap-antennas made of polar materials such as SiC. Finally we use the quantum corrected model (QCM) to show that the morphology of the gap has a dramatic influence on the plasmonic response also for subnanometer gaps. This effect can be crucial for the correct interpretation of charge transfer processes in metallic cavities where quantum effects such as electron tunneling are important.