Eye gaze estimation is increasingly demanded by recent intelligent systems to facilitate a range of interactive applications. Unfortunately, learning the highly complicated regression from a single eye image to the gaze direction is not trivial. Thus, the problem is yet to be solved efficiently. Inspired by the two-eye asymmetry as two eyes of the same person may appear uneven, we propose the face-based asymmetric regression-evaluation network (FARE-Net) to optimize the gaze estimation results by considering the difference between left and right eyes. The proposed method includes one face-based asymmetric regression network (FAR-Net) and one evaluation network (E-Net). The FAR-Net predicts 3D gaze directions for both eyes and is trained with the asymmetric mechanism, which asymmetrically weights and sums the loss generated by two-eye gaze directions. With the asymmetric mechanism, the FAR-Net utilizes the eyes that can achieve high performance to optimize network. The E-Net learns the reliabilities of two eyes to balance the learning of the asymmetric mechanism and symmetric mechanism. Our FARE-Net achieves leading performances on MPIIGaze, EyeDiap and RT-Gene datasets. Additionally, we investigate the effectiveness of FARE-Net by analyzing the distribution of errors and ablation study.