In pose estimation for objects with rotational symmetry, ambiguous poses may arise, and the symmetry axes of objects are crucial for eliminating such ambiguities. Currently, in pose estimation, reliance on manual settings of symmetry axes decreases the accuracy of pose estimation. To address this issue, this method proposes determining the orders of symmetry axes and angles between axes based on a given rotational symmetry type or polyhedron, reducing the need for manual settings of symmetry axes. Subsequently, two key axes with the highest orders are defined and localized, then three orthogonal axes are generated based on key axes, while each symmetry axis can be computed utilizing orthogonal axes. Compared to localizing symmetry axes one by one, the key-axis-based symmetry axis localization is more efficient. To support geometric and texture symmetry, the method utilizes the ADI metric for key axis localization in geometrically symmetric objects and proposes a novel metric, ADI-C, for objects with texture symmetry. Experimental results on the LM-O and HB datasets demonstrate a 9.80% reduction in symmetry axis localization error and a 1.64% improvement in pose estimation accuracy. Additionally, the method introduces a new dataset, DSRSTO, to illustrate its performance across seven types of geometrically and texturally symmetric objects. The GitHub link for the open-source tool based on this method is https://github.com/WangYuLin-SEU/KASAL.