We study the extension of total variation (TV), total deformation (TD), and (second-order) total generalized variation (TGV) to symmetric tensor fields. We show that for a suitable choice of finite-dimensional norm, these variational seminorms are rotation-invariant in a sense natural and well suited for application to diffusion tensor imaging (DTI). Combined with a positive definiteness constraint, we employ these novel seminorms as regularizers in Rudin--Osher--Fatemi (ROF) type denoising of medical in vivo brain images. For the numerical realization, we employ the Chambolle--Pock algorithm, for which we develop a novel duality-based stopping criterion which guarantees error bounds with respect to the functional values. Our findings indicate that TD and TGV, both of which employ the symmetrized differential, provide improved results compared to other evaluated approaches.