  We study the theory of safety and liveness in a reversible calculus where
reductions are totally ordered and rollbacks lead the systems to past states.
Similar to previous work on communicating transactions, liveness and safety
respectively correspond to the should-testing and inverse may-testing
preorders. We develop fully abstract models for these preorders in a reversible
calculus, which are based only on forward transitions, thus providing a simple
proof technique for refinement of such systems. We show that with respect to
safety, total reversibility is a conservative extension to CCS. With respect to
liveness, however, adding total reversibility to CCS distinguishes more
systems. To our knowledge, this work provides the first characterisations of
safety and liveness, and the first testing theory for a reversible calculus.
