Image registration under domain shift remains a fundamental challenge in computer vision and medical imaging: when source and target images exhibit systematic intensity differences, the brightness constancy assumption underlying conventional registration methods is violated, rendering correspondence estimation ill-posed. We propose SAR-Net, a unified framework that addresses this challenge through principled scene-appearance disentanglement. Our key insight is that observed images can be decomposed into domain-invariant scene representations and domain-specific appearance codes, enabling registration via re-rendering rather than direct intensity matching. We establish theoretical conditions under which this decomposition enables consistent cross-domain alignment (Proposition 1) and prove that our scene consistency loss provides a sufficient condition for geometric correspondence in the shared latent space (Proposition 2). Empirically, we validate SAR-Net on the ANHIR (Automatic Non-rigid Histological Image Registration) challenge benchmark, where multi-stain histopathology images exhibit coupled domain shift from different staining protocols and geometric distortion from tissue preparation. Our method achieves a median relative Target Registration Error (rTRE) of 0.25%, outperforming the state-of-the-art MEVIS method (0.27% rTRE) by 7.4%, with robustness of 99.1%. Code is available at https://github.com/D-ST-Sword/SAR-NET .