Cross-domain HVAC energy prediction is essential for scalable building energy management, particularly because collecting extensive labeled data for every new building is both costly and impractical. Yet, this task remains highly challenging due to the scarcity and heterogeneity of data across different buildings, climate zones, and seasonal patterns. In particular, buildings situated in distinct climatic regions introduce variability that often leads existing methods to overfit to spurious correlations, rely heavily on expert intervention, or compromise on data diversity. To address these limitations, we propose CaberNet, a causal and interpretable deep sequence model that learns invariant (Markov blanket) representations for robust cross-domain prediction. In a purely data-driven fashion and without requiring any prior knowledge, CaberNet integrates i) a global feature gate trained with a self-supervised Bernoulli regularization to distinguish superior causal features from inferior ones, and ii) a domain-wise training scheme that balances domain contributions, minimizes cross-domain loss variance, and promotes latent factor independence. We evaluate CaberNet on real-world datasets collected from three buildings located in three climatically diverse cities, and it consistently outperforms all baselines, achieving a 22.9% reduction in normalized mean squared error (NMSE) compared to the best benchmark. Our code is available at https://github.com/SusCom-Lab/CaberNet-CRL.