There are quite a few results in the study of computable linear orders of the form $\tau \cdot L$ is computable iff $L$ is $0^{(n)}$-computable". The goal of joint work with Frolov, Ng and Wu is to classify the order types $\tau$ for which such statements are true. We will concentrate on the case $n=0$, where we have an exact classication: An order type $\tau$ has the property that $\tau \cdot L$ is computable iff $L$ is computable, iff $\tau$ is finite and nonempty. I will conclude with some general comments.