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Extensive studies on Unsupervised Domain Adaptation (UDA) have propelled the deployment of deep learning from limited experimental datasets into real-world unconstrained domains. Most UDA approaches align features within a common embedding space and apply a shared classifier for target prediction. However, since a perfectly aligned feature space may not exist when the domain discrepancy is large, these methods suffer from two limitations. First, the coercive domain alignment deteriorates target domain discriminability due to lacking target label supervision. Second, the source-supervised classifier is inevitably biased to source data, thus it may underperform in target domain. To alleviate these issues, we propose to simultaneously conduct feature alignment in two individual spaces focusing on different domains, and create for each space a domain-oriented classifier tailored specifically for that domain. Specifically, we design a Domain-Oriented Transformer (DOT) that has two individual classification tokens to learn different domain-oriented representations, and two classifiers to preserve domain-wise discriminability. Theoretical guaranteed contrastive-based alignment and the source-guided pseudo-label refinement strategy are utilized to explore both domain-invariant and specific information. Comprehensive experiments validate that our method achieves state-of-the-art on several benchmarks.