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Cycle reconstruction regularized adversarial training -- e.g., CycleGAN, DiscoGAN, and DualGAN -- has been widely used for image style transfer with unpaired training data. Several recent works, however, have shown that local distortions are frequent, and structural consistency cannot be guaranteed. Targeting this issue, prior works usually relied on additional segmentation or consistent feature extraction steps that are task-specific. To counter this, this work aims to learn a general add-on structural feature extractor, by explicitly enforcing the structural alignment between an input and its synthesized image. Specifically, we propose a novel input-output image patches self-training scheme to achieve a disentanglement of underlying anatomical structures and imaging modalities. The translator and structure encoder are updated, following an alternating training protocol. In addition, the information w.r.t. imaging modality can be eliminated with an asymmetric adversarial game. We train, validate, and test our network on 1,768, 416, and 1,560 unpaired subject-independent slices of tagged and cine magnetic resonance imaging from a total of twenty healthy subjects, respectively, demonstrating superior performance over competing methods.