Is the time reversal symmetry of non-deterministic computations important?
The time reversal symmetry is already important for decision problems. It translates into reversal of the accepted language. The trick is to replace total functions by partial functions, and don't consider "yes"/"no" as output of the problem. Instead, the "yes"/"no" is derived from the existence of a path between start and end state. This is explained for finite automata in my answer to the question "Why is non-determinism useful concept?", and elaborated for more powerful machines like pushdown automata, Turing machines and probabilistic machines in my answer to the question "What is the difference between non-determinism and randomness?".

This raises the question whether deterministic context-free languages are closed under the reversal of $L$, because this is the only one of the examples where the deterministic and the non-deterministic languages are not identical.

The importance of the time reversal symmetry is not necessarily limited to the decision problem. It might be possible to also work out its consequences for the counting problem and the function problem.