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Multi-contrast MRI (MC-MRI) captures multiple complementary imaging modalities to aid in radiological decision-making. Given the need for lowering the time cost of multiple acquisitions, current deep accelerated MRI reconstruction networks focus on exploiting the redundancy between multiple contrasts. However, existing works are largely supervised with paired data and/or prohibitively expensive fully-sampled MRI sequences. Further, reconstruction networks typically rely on convolutional architectures which are limited in their capacity to model long-range interactions and may lead to suboptimal recovery of fine anatomical detail. To these ends, we present a dual-domain self-supervised transformer (DSFormer) for accelerated MC-MRI reconstruction. DSFormer develops a deep conditional cascade transformer (DCCT) consisting of several cascaded Swin transformer reconstruction networks (SwinRN) trained under two deep conditioning strategies to enable MC-MRI information sharing. We further present a dual-domain (image and k-space) self-supervised learning strategy for DCCT to alleviate the costs of acquiring fully sampled training data. DSFormer generates high-fidelity reconstructions which experimentally outperform current fully-supervised baselines. Moreover, we find that DSFormer achieves nearly the same performance when trained either with full supervision or with our proposed dual-domain self-supervision.