Contemporary glioma diagnosis integrates molecular features with histopathology to guide clinical decision-making. However, in clinical settings, divergent imaging protocols result in incomplete MRI sequences, leading to two primary challenges: forcing existing frameworks to discard a large portion of clinical data during training and consequently limiting their clinical applicability. To address these limitations, we propose GMENet, a Generative Mixture of Experts Network for multi-center glioma diagnosis with incomplete imaging sequences. Firstly, we design a Cross-attention-based Gated Generation Module that synthesizes missing sequence features from available sequences via cross-attention and dynamic gating mechanisms, incorporating a cycle-consistency loss to preserve semantic integrity. Secondly, we introduce a Dynamically Weighted Experts Fusion Module that performs mixture-of-experts interaction and confidence-aware fusion over original and synthesized dual-sequence features for multi-task prediction. We evaluate GMENet on a multi-center cohort of 1,241 subjects from four in-house datasets and two public repositories. Experiments show that GMENet expands clinically usable training data by 97%, relative to complete-sequence-only data. Furthermore, it consistently outperforms state-of-the-art methods trained on complete data, demonstrating improved robustness under cross-center distribution shifts. Code is available at: https://github.com/spf-sd/GMENet.