  We continue the study of computable embeddings for pairs of structures, i.e.
for classes containing precisely two non-isomorphic structures. Surprisingly,
even for some pairs of simple linear orders, computable embeddings induce a
non-trivial degree structure. Our main result shows that although $\{\omega
\cdot 2, \omega^\star \cdot 2\}$ is computably embeddable in $\{\omega^2,
{(\omega^2)}^\star\}$, the class $\{\omega \cdot k,\omega^\star \cdot k\}$ is
\emph{not} computably embeddable in $\{\omega^2, {(\omega^2)}^\star\}$ for any
natural number $k \geq 3$.
