Diffeomorphic deformable image registration is crucial in many medical image studies, as it offers unique, special features including topology preservation and invertibility of the transformation. Recent deep learning-based deformable image registration methods achieve fast image registration by leveraging a convolutional neural network (CNN) to learn the spatial transformation from the synthetic ground truth or the similarity metric. However, these approaches often ignore the topology preservation of the transformation and the smoothness of the transformation which is enforced by a global smoothing energy function alone. Moreover, deep learning-based approaches often estimate the displacement field directly, which cannot guarantee the existence of the inverse transformation. In this paper, we present a novel, efficient unsupervised symmetric image registration method which maximizes the similarity between images within the space of diffeomorphic maps and estimates both forward and inverse transformations simultaneously. We evaluate our method on 3D image registration with a large scale brain image dataset. Our method achieves state-of-the-art registration accuracy and running time while maintaining desirable diffeomorphic properties.