I must confess I’m having some trouble figuring out precisely what the puzzle is asking.
I guess it is saying that there are automorphisms of the linked-keyring system that map any pair of rings to any other pair, but this is not true of triples ... but are these automorphisms purely topological bijections on the complement of the rings in R^3, or is there any geometric restriction as well? The bit about not telling left from right suggests that reflections would be valid maps, but I’m a bit nervous about the rings having a specific size and a rigid shape, as if that makes some kind of difference from the case where they were just rubber bands. Or maybe that’s a red herring.