  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
$\omega + 1\leq \alpha$ or $\alpha =\omega$ and $\beta < \omega \beta$.
