This paper deals with parametric techniques for the description of texture on very high resolution (VHR) remote sensing images. These techniques focus on the property of anisotropy as described by the local structure tensor (LST). The novelty of this paper consists in proposing several comprehensive statistical frameworks to handle LST fields for rotation-invariant texture discrimination tasks. These frameworks are all based on probability models defined on the Riemannian manifold of positive definite matrices: a recent Riemannian Gaussian model on the affine-invariant metric space and a multivariate Gaussian distribution on the Log-Euclidean space. A thorough comparison of the proposed methods is performed with respect to some state-of-the-art texture analysis methods. Three experimental protocols are considered based on VHR remote sensing data. The first one consists of a content-based image retrieval (CBIR) protocol for browsing oyster field patches. The second one concerns a supervised classification protocol for grouping maritime pine forest stands in different age classes. The third one is, again, a CBIR protocol performed on the UC Merced land use/land cover patch collection. Tensor-based approaches show similar or even better results than the state-of-the-art texture analysis methods considered for comparison in all the experimental contexts.