Asymmetric rolling texture of aluminium alloy 1050 was examined both experimentally and numerically. The rolling asymmetry was realized using rolls with different diameters rotating with the same angular velocity and by varying inclination of the rolling strip (i.e., the flat and tilted entry of the rolled strip between rolls). The final, 84% reduction in the thickness was obtained after six consecutive rolling passes. Crystallographic texture variation over the rolled bar thickness was determined using X-ray diffraction and predicted using the Finite Element Method combined with two crystalline deformation models (i.e., the elasto-plastic and the elasto-viscoplastic ones). Textures predicted by both models, taking into account all deformation process parameters, are in good agreement with experimental results. The obtained results confirm the effect of texture modifications caused by the shear stress component, resulting in the shifts of selected texture maxima in the orientation space, and explain the observed texture distribution across the sample depth. The novelty of this work consist in examination of the role of the tilted material entry (besides a difference in rolls diameters) in homogenization of texture distribution in multi-pass asymmetric rolling. The rolling geometry process variants, recommended for the technological practice, are indicated.