Imagine a pair of denim pants, one could say it has the property of being made of denim, and when you bend the pants or move them around the room, they keep being made of denim. Anything you do to the pants that you can undo, like moving them around or bending them, those are invertible symmetries. Now imagine I rip the pants in half: they're still made of denim, but this action is not so easy to invert. So there are actions I can perform on these pants that are hard or impossible to undo, but still maintain the property of them being made of denim, these are non-invertible symmetries.
The example given in the article is a system being split into a superposition in a way that's hard or impossible to undo, sort of like the pants being torn in half, but just because the system is now in a superposition doesn't mean there weren't properties preserved along the way.