So what are the implications of an object that breaks that symmetry? Would it be possible to observe an object in different times in the same state, or the same time in different states?
The object would be in a repeating set of states. First state 1, then state 2, then state 1, then back to state 2, etc.
This is not in itself surprising. Think of a pendulum, for instance. The difference here is that the motion is the lowest energy state. Over time, a pendulum swing decays. This does not: it will continue forever, if not disturbed.
The obvious next question is: isn't that perpetual motion? According to the normal dictionary definition of those words yes it is, which is why this is a fascinating discovery. However, it doesn't violate the normal arguments against perpetual motion, because there's no way to extract energy from the system. Anything you did to influence the motion would require adding energy.