Lower bound for orienting an asynchronous ring?
Edit: The answer below was written assuming that the ring is synchronous.

Note that if node ids are chosen from some countable set and you don't care about time complexity, the $\Omega(n\log n)$ messages lower bound for electing a leader in a synchronous ring does not apply.

In that case, there's an $O(n)$ messages algorithm that solves your problem: First, elect a leader using the $O(n)$ time-slicing algorithm of [1], and then, as mentioned in your comment, use the leader to orient the ring. Moreover, $\Omega(n)$ seems to be a trivial lower bound: If $o(n)$ messages are being sent, then there is a segment of $2$ neighboring nodes that do not send/receive any messages throughout the run. By an indistinguishability argument, you can show that there is a run where you get conflicting orientations.

[1] Greg N. Frederickson, Nancy A. Lynch: Electing a leader in a synchronous ring. J. ACM 34(1): 98-115 (1987)