Functional magnetic resonance imaging (fMRI) holds great potential for diagnosing and understanding brain disorders. However, the complexity and subtlety of disease-relevant variations in fMRI present significant challenges. To address this issue, we propose a framework that combines equivariant learning and contrastive learning (ECL) to disentangle disease-relevant patterns from irrelevant patterns in fMRI. The framework uses a personalized mask to separate the functional connectivity network from fMRI into a disease-relevant subgraph and an irrelevant subgraph. The disease-relevant subgraph undergoes an equivariant learning pipeline to align the orbit of the encoded features with the orbit of the augmented views of the inputs. The disease-irrelevant subgraph undergoes a contrastive learning pipeline that pulls the encoded features to be close from augmented views of the same input. By combining these 2 learning processes, the learned encoder can be invariant to perturbations to disease-irrelevant patterns while equivariant to disease-relevant variations. The proposed approach achieved state-of-the-art classification performance across 3 benchmark datasets: ABIDE I, ABIDE II, and ADHD-200, with significant improvements in accuracy (improved by up to 5%). Interpretability experiments identified disease-related regions of interest (ROIs) of clinical relevance. These results establish our framework as a promising tool for analyzing brain networks in fMRI. The code is available at https://github.com/CXshen468/ecl .