Tensegrity structures, which are made of struts and tendons, are attracting attention as a platform for adaptive and resilient robots with connections to biological systems. However, they are difficult to control because of their elasticity, deformability, and tight coupling between their elements. Studies on morphological computation and physical reservoir computing suggest, however, that the temporal patterns generated by body dynamics can be exploited to perform computations. In this work, we analyze the diverse collection of behaviors generated by driving tensegrity robots with simple periodic motor commands. We find that characteristic locomotion gaits, such as sliding and rolling, appear in specific regions of the system parameter space. Furthermore, the analysis shows that both normal and anomalous deterministic diffusion emerge because of interactions of the body with the environment. The highly nonlinear relationship between the parameters and robot behavior highlights the difficulty of controlling tensegrities. However, we demonstrate that our results of these nontrivial relationships can in fact be directly exploited to achieve adaptive behavioral switching. These results point to potential uses of tensegrity dynamics as computational resources.