The demand and usage of 360$^{\circ}$ video services are expected to increase. However, despite these services being highly bandwidth intensive, not much is known about the potential value that basic bandwidth saving techniques such as server or edge-network on-demand caching (e.g., in a CDN) could have when used for delivery of such services. This problem is both important and complicated as client-side solutions have been developed that split the full 360$^{\circ}$ view into multiple tiles, and adapt the quality of the downloaded tiles based on the user's expected viewing direction and bandwidth conditions. This paper presents new trace-based analysis methods that incorporate users' viewports (the area of the full 360$^{\circ}$ view the user actually sees), a first characterization of the cross-user similarities of the users' viewports, and a trace-based analysis of the potential bandwidth savings that caching-based techniques may offer under different conditions. Our analysis takes into account differences in the time granularity over which viewport overlaps can be beneficial for resource saving techniques, compares and contrasts differences between video categories, and accounts for uncertainties in the network conditions and the prediction of the future viewing direction when prefetching. The results provide substantial insight into the conditions under which overlap can be considerable and caching effective, and inform the design of new caching system policies tailored for 360$^{\circ}$ video.