In this paper, we present Iterative Symmetry Completion Network (ISCNet), a single depth-image shape completion method that exploits reflective symmetry cues to obtain more detailed shapes. The efficacy of single depth-image shape completion methods is often sensitive to the accuracy of the symmetry plane. ISCNet therefore jointly estimates the symmetry plane and shape completion iteratively; more complete shapes contribute to more robust symmetry plane estimates and vice versa. Furthermore, our shape completion method operates in the image domain, enabling more efficient high-resolution, detailed geometry reconstruction. We perform the shape completion from pairs of viewpoints, reflected across the symmetry plane, predicted by a reinforcement learning agent to improve robustness and to simultaneously explicitly leverage symmetry. We demonstrate the effectiveness of ISCNet on a variety of object categories on both synthetic and real-scanned datasets.