Electroencephalography (EEG) is widely used for mental stress classification, but effective feature extraction and transfer across subjects remain challenging due to its variability. In this paper, a novel deep neural network combining convolutional neural network (CNN) and adversarial theory, named symmetric deep convolutional adversarial network (SDCAN), is proposed for stress classification based on EEG. The adversarial inference is introduced to automatically capture invariant and discriminative features from raw EEG, which aims to improve the classification accuracy and generalization ability across subjects. Experiments were conducted with 22 human subjects, where each participant's stress was induced by the Trier Social Stress Test paradigm while EEG was collected. Stress states were then calibrated into four or five stages according to the changing trend of salivary cortisol concentration. The results show that the proposed network achieves improved accuracies of 87.62% and 81.45% on the classification of four and five stages, respectively, compared to conventional CNN methods. Euclidean space data alignment approach (EA) was applied and the improved generalization ability of EA-SDCAN across subjects was also validated via the leave-one-subject-out-cross-validation, with the accuracies of four and five stages being 60.52% and 48.17%, respectively. These findings indicate that the proposed SDCAN network is more feasible and effective for classifying the stages of mental stress based on EEG compared with other conventional methods.