@Dynamically Linked:

Interesting point, though I can't believe Barbour et. al. wouldn't have thought of it. My own first reaction is, "You can detect left-right reversals relative to the rest of the universe, but reversing left and right of the entire universe would give you the same internal experience." It would also reverse the meaning of the words "left" and "right" - you have to be careful not to imagine yourself looking at the universe from outside.

@Recovering: I expect most physicists have never heard of relative configuration spaces as such. But e.g. Lee Smolin is a coauthor of Barbour's.