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This work systematically investigates the texture-property linkages in hexagonal close-packed (hexagonal) materials using a three-dimensional computational crystal plasticity approach. Magnesium and its alloys are considered as a model system. We perform full-field, large-strain, micromechanical simulations using a wide range of surrogate textures that also sample several experimental datasets for a range of Mg alloys. The role of textural variability and the associated sensitivity of deformation mechanisms on the evolution of the macroscopic plastic anisotropy and strength asymmetry is mapped under uniaxial tensile and compressive loading along the material principal and off-axes orientations. To assess the role of crystallographic plastic anisotropy, two distinct material datasets are simulated, which represent pure and alloyed magnesium. The results provide insights into experimental observations reported for magnesium alloys over a range of materials textures. We discuss potential implications on the damage tolerance from the aggregate plastic anisotropy arising from intrinsic crystallographic and textural effects.