Robot localization is the problem of determining a robot's pose in an environment, typically within a given map or a similar representation. Different methods have been proposed to address this localization problem for robots with limited sensing. In this letter, we present a localization method for a robot equipped with only a contact sensor and a clock. We make the limits of localization accuracy precise by establishing the fundamental limits imposed by symmetry as revealed by the robot's sensors. Our method is based on finding periodic cycles and transient trajectories of the robot path as it bounces within an environment filled with obstacles. Based on the cycles and transient trajectories, space-efficient and automata-based combinatorial filters are synthesized to solve localization problems modulo symmetries. Experimental results from multiple simulations and from real robot demonstrations attest to the feasibility and practicability of our method.