Image fusion synthesizes complementary information from multiple sources, mitigating the inherent limitations of unimodal imaging systems. Accurate image registration is essential for effective multi-source data fusion. However, existing registration methods, often based on image translation in Euclidean space, fail to handle cross-modal misalignment effectively, resulting in suboptimal alignment and fusion quality. To overcome this limitation, we explore image alignment in non-Euclidean space and propose a Hyperbolic Cycle Alignment Network (Hy-CycleAlign). To the best of our knowledge, Hy-CycleAlign is the first image registration method based on hyperbolic space. It introduces a dual-path cross-modal cyclic registration framework, in which a forward registration network aligns cross-modal inputs, while a backward registration network reconstructs the original image, forming a closed-loop registration structure with geometric consistency. Additionally, we design a Hyperbolic Hierarchy Contrastive Alignment (H$^{2}$CA) module, which maps images into hyperbolic space and imposes registration constraints, effectively reducing interference caused by modality discrepancies. We further analyze image registration in both Euclidean and hyperbolic spaces, demonstrating that hyperbolic space enables more sensitive and effective multi-modal image registration. Extensive experiments on misaligned multi-modal images demonstrate that our method significantly outperforms existing approaches in both image alignment and fusion. Our code will be publicly available.