We propose an algorithm for the diffeomorphic registration of diffusion tensor images (DTI). Previous DTI registration algorithms using full tensor information suffer from difficulties in computing the differential of the Finite Strain tensor reorientation strategy. We borrow results from computer vision to derive an analytical gradient of the objective function. By leveraging on the closed-formgradient and the one-parameter subgroups of diffeomorphisms, the resulting registration algorithm is diffeomorphic and fast. Registration of a pair of 128 × 128 × 60 diffusion tensor volumes takes 15 minutes. We contrast the algorithm with a classic alternative that does not take into account the reorientation in the gradient computation. We show with 40 pairwise DTI registrations that using the exact gradient achieves significantly better registration.