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In 3D action recognition, there exists rich complementary information between skeleton modalities. Nevertheless, how to model and utilize this information remains a challenging problem for self-supervised 3D action representation learning. In this work, we formulate the cross-modal interaction as a bidirectional knowledge distillation problem. Different from classic distillation solutions that transfer the knowledge of a fixed and pre-trained teacher to the student, in this work, the knowledge is continuously updated and bidirectionally distilled between modalities. To this end, we propose a new Cross-modal Mutual Distillation (CMD) framework with the following designs. On the one hand, the neighboring similarity distribution is introduced to model the knowledge learned in each modality, where the relational information is naturally suitable for the contrastive frameworks. On the other hand, asymmetrical configurations are used for teacher and student to stabilize the distillation process and to transfer high-confidence information between modalities. By derivation, we find that the cross-modal positive mining in previous works can be regarded as a degenerated version of our CMD. We perform extensive experiments on NTU RGB+D 60, NTU RGB+D 120, and PKU-MMD II datasets. Our approach outperforms existing self-supervised methods and sets a series of new records. The code is available at https://github.com/maoyunyao/CMD