For my intermediate question about generalizing partial orders to something where order-preserving and order-reversing maps would both be homomorphisms, here is a sort of brute-force way to make a poset forget its orientation:
We consider a four-way relation $ab\sim cd$, intuitively meaning "the relation between $a$ and $b$ is the same as the relation between $c$ and $d$", and formally defined by
$$ ab\sim cd ~\equiv~ (a<b\land c<d)\lor(b<a\land d<c) $$
Celarly, if a partial order $P$ has at least one pair of different but comparable elements, then the $\sim$-preserving maps from $P$ to another partial order are exactly the order-preserving maps plus the order-reversing maps. The same is also (somewhat vacuously) true if $P$ is the trivial partial order.
We can give axioms for $\sim$ that guarantee that it derives from a partial order:
- Outer symmetry: $ab\sim xy ~\Rightarrow~ xy\sim ab$
- Outer transitivity: $ab\sim xy \land xy\sim pq ~\Rightarrow~ ab\sim pq$
- Inner symmetry: $ab\sim xy ~\Rightarrow~ ba\sim yx$
- Inner transitivity: $ab\sim bc ~\Rightarrow~ ab\sim ac$
- Inner irreflexivity: $\neg(ab\sim xx)$
- Connectedness: $ab\sim xy \land pq\sim rs ~\Rightarrow~ ab\sim pq \lor ab\sim qp$
For any $\sim$ that satisfies these axioms we can derive a partial order that induces it: If $\sim$ is the empty relation, then the trivial partial order works. Otherwise choose fixed $a$ and $b$ such that $ab\sim pq$ for some $p$ and $q$ and define $x<y$ to mean $ab\sim xy$. Then $x<y$ is a strict partial order, and $\sim$ is the quaternary relation induced by it.
But elegant it ain't. In particular the connectedness axiom looks rather ad-hoc. (On the other hand, this could be a sign that deleting this axiom might lead to something interesting beyond just posets).
I'm still curious whether something interesting can be done with the "betweenness" idea, though.