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Our contribution is a unified cross-modality feature disentagling approach for multi-domain image translation and multiple organ segmentation. Using CT as the labeled source domain, our approach learns to segment multi-modal (T1-weighted and T2-weighted) MRI having no labeled data. Our approach uses a variational auto-encoder (VAE) to disentangle the image content from style. The VAE constrains the style feature encoding to match a universal prior (Gaussian) that is assumed to span the styles of all the source and target modalities. The extracted image style is converted into a latent style scaling code, which modulates the generator to produce multi-modality images according to the target domain code from the image content features. Finally, we introduce a joint distribution matching discriminator that combines the translated images with task-relevant segmentation probability maps to further constrain and regularize image-to-image (I2I) translations. We performed extensive comparisons to multiple state-of-the-art I2I translation and segmentation methods. Our approach resulted in the lowest average multi-domain image reconstruction error of 1.34$\pm$0.04. Our approach produced an average Dice similarity coefficient (DSC) of 0.85 for T1w and 0.90 for T2w MRI for multi-organ segmentation, which was highly comparable to a fully supervised MRI multi-organ segmentation network (DSC of 0.86 for T1w and 0.90 for T2w MRI).