Pairs of orthogonal countable ordinals

We characterize pairs of orthogonal countable ordinals. Two ordinals $alpha$ and $beta$ are orthogonal if there are two linear orders $A$ and $B$ on the same set $V$ with order types $alpha$ and $beta$ respectively such that the only maps preserving both orders are the constant maps and the identity map. We prove that if $alpha$ and $beta$ are two countable ordinals, with $alpha leq beta$, then $alpha$ and $beta$ are orthogonal if and only if either $omega + 1leq alpha$ or $alpha =omega$ and $beta < omega beta$.