For 3D scatterplots, we present an interpolation and projection technique that supports the smooth exchange of one or two data dimensions at a time. Even though this exchange can be considered as a rotation in 4D or 5D data domains, we guarantee that the projection to image space is perceived as a 3D rigid body rotation--with a consistent motion of the data points. We conducted a controlled user study showing that 3D rigid body rotations outperform direct transition between scatterplots. We further extend our technique to support navigation between 3D scatterplots by introducing 3D scatterplot matrices. The usefulness of our approach is demonstrated by application examples, including a case study with a natural language processing expert.