Pairs of orthogonal countable ordinals

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