This paper challenges the cross-species generalization problem for animal pose estimation, aiming to learn a pose estimator that can be well generalized to novel species. We find the relation between different joints is important with two-fold impact: 1) on the one hand, some relation is consistent across all the species and may help two joints mutually confirm each other, e.g., the eyes help confirm the nose and vice versa because they are close in all species. 2) on the other hand, some relation is inconsistent for different species due to the species variation and may bring severe distraction rather than benefit. With these two insights, we propose a Decompose-to-Generalize (D-Gen) pose estimation method to break the inconsistent relations while preserving the consistent ones. Specifically, D-Gen first decomposes the body joints into several joint concepts so that each concept contains multiple closely-related joints. Given these joint concepts, D-Gen 1) promotes the interaction between intra-concept joints to enhance their reliable mutual confirmation, and 2) suppresses the interaction between inter-concept joints to prohibit their mutual distraction. Importantly, we explore various decomposition approaches, i.e., heuristic, geometric and attention-based approaches. Experimental results show that all these decomposition manners yield reasonable joint concepts and substantially improve cross-species generalization (and the attention-based approach is the best).