Geometric view of class of equivalence of $S^1$

Question

I was faced by this question a while ago and I can't proceed: Define the equivalence relation in $S^1$ as $(a,b)R(x,y)$ if $a-b=x-y$. Now, its two questions, since is quite easy to show that $R$ is a equivalence relation:

1. How can I discribe geometrically the classes of R?
2. Who are the unit classes of R?

This seems easy, but for some reason I can't proceed with this. Any help would be very appreciated! Thanks in advance!