  Work of Linnell shows that the space of left-orderings of a group is either
finite or uncountable, and in the case that the space is finite, the
isomorphism type of the group is known---it is what is known as a Tararin
group. By defining semiconjugacy of circular orderings in a general setting
(that is, for arbitrary circular orderings of groups that may not act on
$S^1$), we can view the subspace of left-orderings of any group as a single
semiconjugacy class of circular orderings. Taking this perspective, we
generalize the result of Linnell, to show that every semiconjugacy class of
circular orderings is either finite or uncountable, and when a semiconjugacy
class is finite, the group has a prescribed structure. We also investigate the
space of left-orderings as a subspace of the space of circular orderings,
addressing a question of Baik and Samperton.
