Carbon nanotubes(1,2) can be distinctly metallic or semiconducting depending on their diameter and chirality(3). Here we show that continuously varying the chirality by mechanical torsion(4) can induce conductance oscillations, which can be attributed to metal - semiconductor periodic transitions. The phenomenon is observed in multiwalled carbon nanotubes, where both the torque(5) and the current are shown to be carried predominantly by the outermost wall(6,7). The oscillation period with torsion is consistent with the theoretical shifting(8) of the corners of the first Brillouin zone of graphene across different subbands allowed in the nanotube. Beyond a critical torsion, the conductance irreversibly drops due to torsional failure, allowing us to determine the torsional strength of carbon nanotubes. Carbon nanotubes could be ideal torsional springs for nanoscopic pendulums(4,9,10), because electromechanical detection of motion could replace the microscopic detection techniques used at present. Our experiments indicate that carbon nanotubes could be used as electronic sensors of torsional motion in nanoelectro-mechanical systems(11).