Nonrigid registration is vital to medical image analysis but remains challenging for diffusion MRI (dMRI) due to its high-dimensional, orientation-dependent nature. While classical methods are accurate, they are computationally demanding, and deep neural networks, though efficient, have been underexplored for nonrigid dMRI registration compared to structural imaging. We present a novel, deep learning framework for model-free, nonrigid registration of raw diffusion MRI data that does not require explicit reorientation. Unlike previous methods relying on derived representations such as diffusion tensors or fiber orientation distribution functions, in our approach, we formulate the registration as an equivariant diffeomorphism of position-and-orientation space. Central to our method is an $\mathsf{SE}(3)$-equivariant UNet that generates velocity fields while preserving the geometric properties of a raw dMRI's domain. We introduce a new loss function based on the maximum mean discrepancy in Fourier space, implicitly matching ensemble average propagators across images. Experimental results on Human Connectome Project dMRI data demonstrate competitive performance compared to state-of-the-art approaches, with the added advantage of bypassing the overhead for estimating derived representations. This work establishes a foundation for data-driven, geometry-aware dMRI registration directly in the acquisition space.