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As real-world prediction systems often face missing modalities at inference, incomplete multimodal learning (IML) remains a practical challenge. While prior methods aim to learn representations robust to missing inputs, representations from incomplete modalities inevitably deviate from their full-modality counterparts due to missing evidence. To explicitly leverage these deviations, we propose MARS (Missingness-Aware Residual-guided Specialization), a mixture-of-experts framework that guides expert specialization based on how representations are reshaped by missingness. By contrasting task representations derived from incomplete inputs with their complete counterparts during training, we derive a privileged residual signal that captures this representational gap. The residual signal guides a residual router to assign samples to experts specialized for the corresponding deviation patterns. In parallel, a feature router learns to imitate this routing behavior using only incomplete inputs, enabling deployment without access to full modalities. To mitigate this train-test router gap, we develop a discrepancy-aware noise regularization that adaptively perturbs the residual router's decisions when the feature router deviates, enhancing expert robustness under imperfect imitation. Experiments on multimodal classification (CASIA-SURF, CREMA-D, UPMC Food-101) and segmentation (MCubeS) under missing scenarios show that MARS consistently surpasses baselines while remaining efficient and extensible to diverse backbones and tasks.