(joint with Franziska Jahnke, Anna de Mase, Pierre Touchard) In 2011, Cluckers and Halupczok, building from work of Gurevitch and Schmitt, showed that the theory of ordered abelian groups (OAG) admit quantifier elimination down to their spines. Building in turn on their work, we discuss "structural" questions about OAG equipped with their spines: not all substructures of an OAG equipped with its spines are themselves OAG equipped with spines. However, convex subgroups are always well behaved in this regard. Continuing our study, we show that all OAG are AI, where AI stands of course for augmentable at infinity: for any OAG ?, there exists an OAG ? such that ? is an elementary substructure of the direct sum ???, lexicographically ordered. In other words, there is an elementary extension of ? containing ? itself as a convex subgroup.