  A rotational subset, relative to a continuous transformation $T: \mathbb{T}
\to \mathbb{T}$ on the unit circle, is a closed, invariant subset of
$\mathbb{T}$ that is minimal and on which $T$ respects the standard orientation
of the unit circle. In the case where $T$ is the standard angle doubling map,
such subsets were studied by Bullet and Sentenac. The case where $T$ multiplies
angles by an integer $d > 2$ was studied by Goldberg and Tresser, and Blokh,
Malaugh, Mayer, Oversteegen, and Parris. These authors prove that infinite
rotational subsets arise as extensions of irrational rotations of the unit
circle. In this paper, we prove that such a structure theorem holds for the
wider class of continuous transformations $T$ with finite fibers. Our methods
are more squarely analytic in nature than the works mentioned, and hence of
interest even in the cases treated by the works mentioned above. The paper
concludes with an exposition of those cases from the point of view taken here.
