It doesn't seem to work well for smaller sizes when shifting left by 1. I'm sitting in a waiting room, playing with this on paper with four-digit binary numbers, and so far it seems to rapidly fall into a loop of 1,3,5,15,1...

edit: they all fall into their own loops. I forget the term for it, but the pattern is, unless I've made an arithmetic mistake (likely!):

1,3,5,15,1...

2,6,10,14,2...

4,12,4...

7,9,11,(1,3,5,15,1...)

8,8...

14,7,9,11,(1,3,5,15,1...)