In this paper, we give a general characterization of regularization functionals for vector field reconstruction, based on the requirement that the said functionals satisfy certain geometric invariance properties with respect to transformations of the coordinate system. In preparation for our general result, we also address some commonalities of invariant regularization in scalar and vector settings, and give a complete account of invariant regularization for scalar fields, before focusing on their main points of difference, which lead to a distinct class of regularization operators in the vector case. Finally, as an illustration of potential, we formulate and compare quadratic (L(2)) and total-variation-type (L(1)) regularized denoising of vector fields in the proposed framework.