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Unsupervised domain adaptation (UDA) has been vastly explored to alleviate domain shifts between source and target domains, by applying a well-performed model in an unlabeled target domain via supervision of a labeled source domain. Recent literature, however, has indicated that the performance is still far from satisfactory in the presence of significant domain shifts. Nonetheless, delineating a few target samples is usually manageable and particularly worthwhile, due to the substantial performance gain. Inspired by this, we aim to develop semi-supervised domain adaptation (SSDA) for medical image segmentation, which is largely underexplored. We, thus, propose to exploit both labeled source and target domain data, in addition to unlabeled target data in a unified manner. Specifically, we present a novel asymmetric co-training (ACT) framework to integrate these subsets and avoid the domination of the source domain data. Following a divide-and-conquer strategy, we explicitly decouple the label supervisions in SSDA into two asymmetric sub-tasks, including semi-supervised learning (SSL) and UDA, and leverage different knowledge from two segmentors to take into account the distinction between the source and target label supervisions. The knowledge learned in the two modules is then adaptively integrated with ACT, by iteratively teaching each other, based on the confidence-aware pseudo-label. In addition, pseudo label noise is well-controlled with an exponential MixUp decay scheme for smooth propagation. Experiments on cross-modality brain tumor MRI segmentation tasks using the BraTS18 database showed, even with limited labeled target samples, ACT yielded marked improvements over UDA and state-of-the-art SSDA methods and approached an "upper bound" of supervised joint training.