We consider the problem of stably separating the orbits of the action of a group of isometries on Euclidean space. We will present two different constructions of maps that can separate the orbits of such an action and discuss when these can also be bilipschitz in an appropriate metric. Both of these constructions can be viewed as generalizations of phase retrieval which we will use as a prototypical example. Time permitting we will also discuss some recent results on optimal bilipschitz embeddings and approximations of such embeddings.