End-to-end weakly supervised semantic segmentation aims at optimizing a segmentation model in a single-stage training process based on only image annotations. Existing methods adopt an online-trained classification branch to provide pseudo annotations for supervising the segmentation branch. However, this strategy makes the classification branch dominate the whole concurrent training process, hindering these two branches from assisting each other. In our work, we treat these two branches equally by viewing them as diverse ways to generate the segmentation map, and add interactions on both their supervision and operation to achieve mutual promotion. For this purpose, a bidirectional supervision mechanism is elaborated to force the consistency between the outputs of these two branches. Thus, the segmentation branch can also give feedback to the classification branch to enhance the quality of localization seeds. Moreover, our method also designs interaction operations between these two branches to exchange their knowledge to assist each other. Experiments indicate our work outperforms existing end-to-end weakly supervised segmentation methods.