Definability of linear orders over negative equivalences

We study linear orders definable over negative and positive equivalences and their computable automorphisms. Special attention is paid to equivalences like $\eta(\alpha)=\alpha^2\cup\mathrm{id}_\omega$, $\alpha\subseteq\omega$. In particular, we describe orders that have negative presentations over such equivalences for co-enumerable sets $\alpha$. Presentable and nonpresentable order types are exemplified for equivalences with various extra properties. We also give examples of negative orders with computable automorphisms whose inverses are not computable.