Consider Jordan arcs on one-dimensional circle, i.e. continuous and injective maps $\varphi: [0, 1] \to \mathbb{S}^1$.
We will say that $\varphi \simeq \psi$ if there is a homeomorphism $f$ of $\mathbb{S}^1$ such that $f$ is isotopic to identity, $(f\circ\varphi)([0, 1]) = \psi([0, 1])$ and $(f \circ \varphi)(0) = \psi(0)$, i.e. homeomorphism which maps one arc to another preserving starting points. Clearly it is an equivalence relation.
I would like to prove that there are exactly two equivalence classes of $\mathrm{Homeo}(\mathbb{S}^1)/\simeq$.
The idea is that all arcs "going in the same direction with respect to starting points" are equivalent, and "going in different directions" are not. Thus, we have exactly two possibilities for any class (clockwise or counterclockwise direction), which finishes the proof. Nevertheless, I do not know how to write a precise proof of this idea.