Let $(X,\leq)$ be a pospace: a topological poset such that uppersets and downsets are closed in the topology. Let $A\subseteq X$ be a closed (connected) subspace and suppose $\phi: A\rightarrow A$ is an order isomorphism: a monotone bijection whose inverse is also monotone. Does $\phi$ extend to an order isomorphism of $X$? If not, what kind of conditions on $X$ do guarantee this property?
I'm specifically interested in the case where $X=[0,1]^n$ in which case things like Tietze's extension theorem might be helpful (or just the general fact that it is a compact metric space).