We study the theory of safety and liveness in a reversible calculus where reductions are totally ordered and rollbacks lead systems to past states. Liveness and safety in this setting naturally correspond to the should-testing and inverse may-testing preorders, respectively. In reversible languages, however, the natural models of these preorders would need to be based on both forward and backward transitions, thus offering complex proof techniques for verification. Here we develop novel fully abstract models of liveness and safety which are based on forward transitions and limited rollback points, giving rise to considerably simpler proof techniques. Moreover, we show that, with respect to safety, total reversibility is a conservative extension to CCS. With respect to liveness, we prove that adding total reversibility to CCS distinguishes more systems. To our knowledge, this work provides the first testing theory for a reversible calculus, and paves the way for a testing theory for causal reversibility.