Tensegrity mobile robots, with their high stiffness-to-mass ratio and exceptional structural adaptability, have become a cutting-edge research focus in recent year. In this paper, a novel spatial multi-locomotion tensegrity mobile robot is put forward. The uniqueness of the robot lies in its ability to achieve three types of spatial locomotion modes in terms of earthworm-like, inchworm-like and turning locomotion, which endows the robot with the ability of smooth directional changing maneuvers. A general kinematic model of the spatial tensegrity robot is established, and motion planning of the three locomotion modes is carried out. On this basis, a prototype of the robot is built, and experimental results show that the tensegrity robot can effectively achieve the three spatial locomotion modes. In particular, benefit from the turning locomotion mode, the robot is capable of executing agile obstacle avoidance maneuvers and performing reverse movements with precision. This work offers valuable insights into the deployment of tensegrity structures for enhancing spatial multi-locomotion capabilities in mobile robotics.