Existing deep learning methods for graph matching(GM) problems usually considered affinity learningto assist combinatorial optimization in a feedforward pipeline, and parameter learning is executed by back-propagating the gradients of the matching loss. Such a pipeline pays little attention to the possible complementary benefit from the optimization layer to the learning component. In this paper, we overcome the above limitation under a deep bidirectional learning framework.Our method circulates the output of the GM optimization layer to fuse with the input for affinity learning. Such direct feedback enhances the input by a feature enrichment and fusion technique, which exploits andintegrates the global matching patterns from the deviation of the similarity permuted by the current matching estimate. As a result, the circulation enables the learning component to benefit from the optimization process, taking advantage of both global feature and the embedding result which is calculated by local propagationthrough node-neighbors. Moreover, circulation consistency induces an unsupervised loss that can be implemented individually or jointly to regularize the supervised loss. Experiments on challenging datasets demonstrate the effectiveness of our methods for both supervised learning and unsupervised learning.