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In recent years, tensegrity-based systems have been gaining increasing attention in the field of mobile robots due to their remarkable characteristics, including high impact tolerance, load-bearing capacity, and inherent self-stabilization capabilities. Despite these advantages, the majority of rolling tensegrity robots relies on center-of-mass shifting for locomotion, which often results in limited agility and reduced efficiency. This article introduces a propulsion-driven spherical tensegrity robot capable of executing multiple rolling gait patterns. Unlike traditional designs, this robot achieves not only the three existing rolling gaits around each side of the landing triangle but also three additional rolling gaits around the vertices of the landing triangle. The proposed multiple-gait rolling mechanism of the novel robot is presented, and the corresponding required driving force for each gait as well as the propulsion model of actuators is analyzed. Then, the performance of the novel robot is evaluated and compared with traditional six-rod spherical tensegrity robots, in aspects of achievable region, path space, step length, cost of transport, and long-distance navigation. Finally, to validate the proposed design scheme, a functional prototype of the tensegrity robot is constructed and subjected to a series of experiments. The results show that the novel robot exhibits superior gait versatility and improves rolling efficiency and maneuverability compared with traditional spherical tensegrity robots.