This work proposes a multimodal diffeomorphic registration method using Neural Ordinary Differential Equations (Neural ODEs). Nonrigid registration algorithms exhibit tradeoffs between their accuracy, the computational complexity of their deformation model, and its proper regularization. In addition, they also assume intensity correlation in anatomically homologous regions of interest among image pairs, limiting their applicability to the monomodal setting. Unlike learning-based models, we propose an instance-specific framework that is not subject to high scan requirements for training and does not suffer performance degradation at inference time on modalities unseen during training. Our method exploits the potential of continuous-depth networks in the Neural ODE paradigm with structural descriptors, widely adopted as modality-agnostic metric models which exploit self-similarities on parameterized neighborhood geometries. We propose three different variants that integrate image-based or feature-based structural descriptors and nonstructural image similarities computed by local mutual information. We conduct extensive evaluations on different experiments formed by scan dataset combinations and show surpassing qualitative and quantitative results compared to state-of-the-art baselines adequate for large or small deformations, and specific of multimodal registration. Lastly, we also demonstrate the underlying robustness of the proposed framework to varying levels of explicit regularization while maintaining low error, its suitability for registration at varying scales, and its efficiency with respect to other methods targeted to large-deformation registration.