Multimodal image registration is a fundamental task and a prerequisite for downstream cross-modal analysis. Despite recent progress in shared feature extraction and multi-scale architectures, two key limitations remain. First, some methods use disentanglement to learn shared features but mainly regularize the shared part, allowing modality-private cues to leak into the shared space. Second, most multi-scale frameworks support only a single transformation type, limiting their applicability when global misalignment and local deformation coexist. To address these issues, we formulate hybrid multimodal registration as jointly learning a stable shared feature space and a unified hybrid transformation. Based on this view, we propose HRNet, a Hybrid Registration Network that couples representation disentanglement with hybrid parameter prediction. A shared backbone with Modality-Specific Batch Normalization (MSBN) extracts multi-scale features, while a Cross-scale Disentanglement and Adaptive Projection (CDAP) module suppresses modality-private cues and projects shared features into a stable subspace for matching. Built on this shared space, a Hybrid Parameter Prediction Module (HPPM) performs non-iterative coarse-to-fine estimation of global rigid parameters and deformation fields, which are fused into a coherent deformation field. Extensive experiments on four multimodal datasets demonstrate state-of-the-art performance on rigid and non-rigid registration tasks. The code is available at the project website.