A homeomorphism $\mathbb R\to\mathbb R$ is almost the same thing as an order isomorphism, except that a homeophorphism can also be an order anti-isomorphism.
I'm wondering whether there is a natural first-order structure "X" which generalizes partial orders (in the sense that order-presering and order-reversing maps would be the prototypical examples of "X morphisms") such that the homeomorphisms $\mathbb R\to\mathbb R$ are exactly the "X isomorphisms".
So far the most promising approach seems to be consider a trinary "betweenness" relation $$\beta(a,b,c) \equiv (a\le b\le c) \lor (c\le b\le a)$$ and look at the category of $\beta$-preserving maps.
Have such structures been studied? Do they have a name? Is there a nice axiomatic characterization of the trinary relations that can be induced by a partial (or total?) order in this way?