The phrase “time reversal symmetry” is very interesting. If it was said in a science fiction novel then you would think that someone was about to travel backwards in time or something to that effect. If it was used is a scientific paper valid questions would probably be asked along the lines of “doesn’t that break the second law of thermodynamics.” The source of this confusion is the idea that the symmetry applies to the whole system when if fact it only applies to a small portion of it. If a red hot metal ball is left in a room it will always cool down, but each individual atom collision is just as valid in reverse provided some things are true. So in this way the system has time reversal symmetry.
When it comes to topological insulators (insulators in the middle but conductors on the outside); for time symmetry no magnetic field can be present in the insulator. In this case the conduction electrons do not scatter and so can be plotted backwards with equal validity. Recently a topological insulator was designed that doesn’t insulate or conduct electrons, but sound. It was produced using aluminium plates, grooved to allow the topological material to exist in rings forming Time invariant symmetry was essential in producing this affect due to the nature of the propagation. Now the hope is that this research may be able to provide sound insulators that aren’t reciprocal and so allow sound to travel but only in one direction.