Pairs of orthogonal countable ordinals

We characterize pairs of orthogonal countable ordinals. Two ordinals αα and ββ are orthogonal if there are two linear orders AA and BB on the same set VV with order types αα and ββ respectively such that the only maps preserving both orders are the constant maps and the identity map. We prove that if αα and ββ are two countable ordinals, with α≤βα≤β, then αα and ββ are orthogonal if and only if either ω+1≤αω+1≤α or α=ωα=ω and β<ωββ<ωβ.