Many image acquisition techniques used in biomedical imaging, material analysis, and structural geology are capable of acquiring 3D solid images. Computational analysis of these images is complex but necessary, since it is difficult for humans to visualize and quantify their detailed 3D content. One of the most common methods to analyze 3D data is to characterize the volumetric texture patterns. Texture analysis generally consists of encoding the local organization of image scales and directions, which can be extremely diverse in 3D. Current state-of-the-art techniques face many challenges when working with 3D solid texture, where most approaches are not able to consistently characterize both scale and directional information. 3D Riesz-wavelets can deal with both properties. One key property of Riesz filterbanks is steerability, which can be used to locally align the filters and compare textures with arbitrary (local) orientations. This paper proposes and compares three novel local alignment criteria for higher-order 3D Riesz-wavelet transforms. The estimations of local texture orientations are based on higher-order extensions of regularized structure tensors. An experimental evaluation of the proposed methods for the classification of synthetic 3D solid textures with alterations (such as rotations and noise) demonstrated the importance of local directional information for robust and accurate solid texture recognition. These alignment methods achieved an accuracy of 0.95 in the rotated data, three times more than the unaligned Riesz descriptor that achieved 0.32. The accuracy obtained is better than all other techniques that are published and tested on the same database.