Face recognition in the infrared (IR) band has become an important supplement to visible light face recognition due to its advantages of independent background light, strong penetration, ability of imaging under harsh environments such as nighttime, rain and fog. However, cross-spectral face recognition (i.e., VIS to IR) is very challenging due to the dramatic difference between the visible light and IR imageries as well as the lack of paired training data. This paper proposes a framework of bidirectional cross-spectral conversion (BCSC-GAN) between the heterogeneous face images, and designs an adaptive weighted fusion mechanism based on information fusion theory. The network reduces the cross-spectral recognition problem into an intra-spectral problem, and improves performance by fusing bidirectional information. Specifically, a face identity retaining module (IRM) is introduced with the ability to preserve identity features, and a new composite loss function is designed to overcome the modal differences caused by different spectral characteristics. Two datasets of TINDERS and CASIA were tested, where performance metrics of FID, recognition rate, equal error rate and normalized distance were compared. Results show that our proposed network is superior than other state-of-the-art methods. Additionally, the proposed rule of Self Adaptive Weighted Fusion (SAWF) is better than the recognition results of the unfused case and other traditional fusion rules that are commonly used, which further justifies the effectiveness and superiority of the proposed bidirectional conversion approach.