This is mostly a checking of my understanding. Currently I am working through the proof of Denjoy's Theorem in Katok and Hasselblatt's Introduction to the Modern Theory of Dynamical Systems, and specifically Lemma 12.1.2 (p. 402).
What this lemma does is it establishes a (finite) sequence of non-intersecting intervals from the orbit of a point by homeomorphism on the circle with irrational rotation number. We have already proven that such a homeomorphism is semi-conjugate to a rotation by this rotation number. K&H construct these intervals for the rotation, and claim that this automatically gives us the intervals for the homeomorphism because, "the claim only involves the ordering of the orbit... Since f is either conjugate or semiconjugate to an irrational rotation, we can thus assume f itself is an irrational rotation."
I wanted to check this, but the only way I could get this result was by the fact that the surjective continuous map we constructed as the semi-conjugacy between the rotation and the homeomorphism was increasing (only strictly in the case of full conjugacy, of course). In this case it was easy, because if $f$ is the homeomorphism, $r$ is the rotation, and $h$ is continuous surjection satisfying $h\circ f = r \circ h$, then the images of the orbit intervals for $f$ under $h$ correspond exactly to (and do not exceed, as is possible without monotonicity) the intervals formed by the orbit of a point under $r$. However, K&H do not mention the monotonicty, and seem to imply that this relation follows directly from the semi-conjugacy.
I guess the thing I'd worry about is if my semi-conjugacy were instead, say, $x^3 + x^2$, which is surjective and continuous but has a little "wiggle" in the middle, and if the two points in an $f$ orbit, call them $y_i,y_j,$ lie on the wiggle you can get it such that $(x_i,x_j) = (h(y_i),h(y,j)) \subset h((y_i,y_j)),$ where $x_i,x_j$ are part of the orbit of the rotation which we found did what we desired. This $h^{-1}((x_i,x_j)) \subset h^{-1}(h((y_i,y_j)))),$ which also contains $(y_i,y_j),$ and I can't see any good way to get $(y_i,y_j)$ inside of $h^{-1}((x_i,x_j))$ to set up the empty intersection with another interval. Of course, this is not a problem with the monotonic map.
More broadly, I want to make sure that this ordering of orbits is not something generally preserved/fully reversed by semi-conjugacy, as implied by K&H, but a special case of this particular semi-conjugacy. Thoughts?
If you look into the Poincaré Classification, so Theorem 11.2.7, they state that the homeomorphism $f$ with irrational rotation number is semiconjugate to the Rotation $R_\tau$ via a noninvertible, continuous and monotone map $h: \mathbb{S}^1 \to \mathbb{S}^1$.
Not sure if this late answer helps you, but I guess that's what you're looking for.