We study the problem of localizing a configuration of points and planes from the collection of point-to-plane distances. This problem models simultaneous localization and mapping from acoustic echoes as well as the & x201C;structure from sound & x201D; approach to microphone localization with unknown sources. In our earlier work we proposed computational methods for localization from point-to-plane distances and noted that such localization suffers from various ambiguities beyond the usual rigid body motions; in this paper we provide a complete characterization of uniqueness. We enumerate all cases of configurations which lead to the same distance measurements as a function of the number of planes and points, and algebraically characterize the related transformations in both 2D and 3D.