Reconstructing compositional 3D representations of scenes, where each object is represented with its own 3D model, is a highly desirable capability in robotics and augmented reality. However, most existing methods rely heavily on strong appearance priors for object discovery, therefore only working on those classes of objects on which the method has been trained, or do not allow for object manipulation, which is necessary to scan objects fully and to guide object discovery in challenging scenarios. We address these limitations with a novel interaction-guided and class-agnostic method based on object displacements that allows a user to move around a scene with an RGB-D camera, hold up objects, and finally outputs one 3D model per held-up object. Our main contribution to this end is a novel approach to detecting user-object interactions and extracting the masks of manipulated objects. On a custom-captured dataset, our pipeline discovers manipulated objects with 78.3% precision at 100% recall and reconstructs them with a mean chamfer distance of 0.90cm. Compared to Co-Fusion, the only comparable interaction-based and class-agnostic baseline, this corresponds to a reduction in chamfer distance of 73% while detecting 99% fewer false positives.