Reliable decoding of motor intention from high-density surface electromyography (HD-EMG) signals is essential for applications in neural-machine interfaces (NMI). However, the decoding accuracy of HD-EMG is susceptible to cross-session electrode displacements, which can lead to significant shifts in signal distribution. To solve the problem, we proposed a multi-task co-calibration framework to achieve cross-session robust decoding by exploiting the intrinsic correlation between hand gesture classification and continuous joint angle estimation. In the initial training phase, a common feature extractor is trained based on a dual-task architecture using gesture labels and joint angles. In subsequent sessions, only gesture labels are required to complete calibration without repeating angle acquisition. We developed two distinct models: the Parameter Transfer Calibration Network (PTC-Net) fine-tunes feature extractor parameters through gesture labels, and the Generative Collaborative Calibration Network (GCC-Net) uses a conditional adversarial generator network (cGAN) to generate pseudo-labels for regression calibration. Experimental results demonstrated that both models significantly improved decoding performance under electrode displacement scenarios, with classification accuracy increasing by 25.7% on average and joint angle regression error reduced by approximately 42.1%. PTC-Net provides a fast and lightweight calibration strategy, while GCC-Net shows better adaptability in more challenging situations with limited labels and noisy signals. This study provides a scalable and practical solution for decoding HD-EMG signals across sessions with electrode displacement and lays the foundation for building more robust and user-friendly EMG-based control systems.