In previous work we studied left-invariant diffusion on the 2D Euclidean motion group for crossing-preserving coherence-enhancing diffusion on 2D images. In this paper we study the equivalent three-dimensional case. This is particularly useful for processing High Angular Resolution Diffusion Imaging (HARDI) data, which can be considered as 3D orientation scores directly. A complicating factor in 3D is that all practical 3D orientation scores are functions on a coset space of the 3D Euclidean motion group instead of on the entire group. We show that, conceptually, we can still apply operations on the entire group by requiring the operations to be
. Subsequently, we propose to describe the local structure of the 3D orientation score using left-invariant derivatives and we smooth 3D orientation scores using left-invariant diffusion. Finally, we show a number of results for linear diffusion on artificial HARDI data.