Connected nice topological spaces that are transitive but not 2-transitive under homeomorphisms
Let $X=\big(\omega_1\times[0,1)\big)\setminus\{\langle 0,0\rangle\}$ ordered lexicographically. Clearly $X$ is connected and Hausdorff; indeed, $X$ is even hereditarily collectionwise normal. For each $x\in X$ the set $(\leftarrow,x)$ is homeomorphic to $\Bbb R$ with the Euclidean topology, so $X$ is transitive under autohomeomorphisms, but it’s clear that no autohomeomorphism of $X$ can reverse the order of a pair of points: that would require embedding $\omega_1^*$ in an initial segment of $X$.