Let $(X,<)$ be a complete DLO without endpoints, whereby completeness, I mean Dedekind completeness. Let the topology on $X$ be the standard one induced by the open intervals. Then, I believe that $(X,<)\simeq (\mathbb{R},<)$ in the sense of homeomorphisms. If $f$ is the homeomorphism between them, is $f$ necessarily an order isomorphism or completely order-reversing?
The motivation for this question comes from trying to view Souslin's Problem as a topological problem. It asks whether or not such $(X,<)$ is order isomorphic to $(\mathbb{R},<)$. I know that the resolution of this problem is independent of $\mathsf{ZFC}$, but I fail to grasp why this problem is topological in nature unless there is some kind of correspondence between order isomorphicity and homeomorphicity in this situation. Am I missing something obvious?