Initial segments of computable linear orders with additional computable predicates

We study computable linear orders with computable neighborhood and block predicates. In particular, it is proved that there exists a computable linear order with a computable neighborhood predicate, having a $\Pi^0_1$-initial segment which is isomorphic to no computable order with a computable neighborhood predicate. On the other hand, every $\Sigma^0_1$-initial segment of such an order has a computable copy enjoying a computable neighborhood predicate.
Similar results are stated for computable linear orders with a computable block predicate replacing a neighborhood relation. Moreover, using the results obtained, we give a simpler proof for the Coles–Downey–Khoussainov theorem on the existence of a computable linear order with $\Pi^0_2$-initial segment, not having a computable copy.