Summary: Reversible computing usually focuses on how to establish a valid equivalence between the global reversibility and local reversibility in computational systems. Hitherto the equivalence has been precisely developed in cellular automata, combinational circuits and quantum computers, but it implicitly assumes that the underlying systems are synchronously timed. Alternative systems include the delay-insensitive (DI) circuits and asynchronous cellular automata (ACAs), where the local operations of each component (cell) may be executed independently at random times. Despite the randomness associated with asynchronous timings, equivalence between the global and local reversibility can be simply achieved in both DI-circuits [K. Morita, Lect. Notes Comput. Sci. 2055, 102–113 (2001; Zbl 0984.68512)] and ACAs [the first author et al., Lect. Notes Comput. Sci. 2509, 220–229 (2002; Zbl 1029.68101)], provided that their local operations (transitions) are thoroughly serialized. The complete exclusion of concurrency in local behavior, however, will profoundly depress the parallel processing efficiency of DI-circuits as well as the intrinsic massive parallelism of ACAs. This paper aims at exploring what kind of complex global behavior may arise from the concurrent operations that are invertible at local level. To this end, we show that DI-circuits composed of reversible elements can actually exhibit universal input and output behavior, with the universality emerging from the concurrency in reversible local operations. Likewise, by further embedding all circuits into the cellular space, the emergence of universal global transitions from reversible local transitions can be exactly identified in an ACA which, due to the bijectivity of its local function, has significantly lower complexity as compared to other models.