Palmprint recognition has been extensively studied as an effective biometric technique for personal identification. With the rapid development of deep neural networks (DNNs), palmprint recognition methods have achieved remarkable progress. However, their performance often deteriorates significantly under domain shifts. Moreover, existing unsupervised domain adaptation approaches for palmprint recognition typically suffer from unstable training and imprecise feature alignment, thereby limiting their effectiveness. To address these challenges, we propose SPA, a Stable and Precise Alignment framework for cross-domain palmprint recognition. Specifically, we design a lightweight yet robust Style Transformation Module (STM) to mitigate variations in style, color, and illumination. With the aid of STM, we further align joint feature distributions across all high-level layers, achieving more accurate feature alignment and enhancing recognition robustness. We conduct extensive experiments on two public multi-domain palmprint databases encompassing 42 cross-domain scenarios. The results demonstrate that SPA consistently delivers superior performance across both databases, achieving higher recognition accuracy with lower computational overhead compared to existing methods. In particular, SPA improves the average identification accuracies to 94.21% and 81.93%, while reducing the average equal error rates (EER) to 1.36% and 3.62% on the two databases, respectively.