Cross-corpus speech emotion recognition (SER) seeks to generalize the ability of inferring speech emotion from a well-labeled corpus to an unlabeled one, which is a rather challenging task due to the significant discrepancy between two corpora. Existing methods, typically based on unsupervised domain adaptation (UDA), struggle to learn corpus-invariant features by global distribution alignment, but unfortunately, the resulting features are mixed with corpus-specific features or not class-discriminative. To tackle these challenges, we propose a novel Emotion Decoupling aNd Alignment learning framework (EMO-DNA) for cross-corpus SER, a novel UDA method to learn emotion-relevant corpus-invariant features. The novelties of EMO-DNA are two-fold: contrastive emotion decoupling and dual-level emotion alignment. On one hand, our contrastive emotion decoupling achieves decoupling learning via a contrastive decoupling loss to strengthen the separability of emotion-relevant features from corpus-specific ones. On the other hand, our dual-level emotion alignment introduces an adaptive threshold pseudo-labeling to select confident target samples for class-level alignment, and performs corpus-level alignment to jointly guide model for learning class-discriminative corpus-invariant features across corpora. Extensive experimental results demonstrate the superior performance of EMO-DNA over the state-of-the-art methods in several cross-corpus scenarios. Source code is available at https://github.com/Jiaxin-Ye/Emo-DNA.