In General Relativity, one is interested in ?asymptotically Euclidean? Riemannian manifolds, that is, manifolds that look almost like Euclidean space outside some compact set. For such manifolds ? typically accompanied by additional structure such as a (0,2)-tensor field and called ?initial data sets? ?, one is interested in understanding asymptotic geometric invariants such as their ?mass?, ?angular momentum?, and ?center of mass?. To study the latter two, one usually assumes the existence of so-called ?Regge--Teitelboim coordinates? on the asymptotic ?end? of the manifold. We will give examples of asymptotically Euclidean initial data sets which do not possess any Regge--Teitelboim coordinates. We will also show that (asymptotic) harmonic coordinates can be used as a tool in checking whether a given asymptotically Euclidean initial data set possesses Regge--Teitelboim coordinates. This is joint work with Melanie Graf and Jan Metzger. We will also explain the consequences these findings have for the definition of the center of mass, relying on joint work with Nerz and with Sakovich.