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Global localization is essential for robot navigation, of which the first step is to retrieve a query from the map database. This problem is called place recognition. In recent years, LiDAR scan based place recognition has drawn attention as it is robust against the appearance change. In this paper, we propose a LiDAR-based place recognition method, named Differentiable Scan Context with Orientation (DiSCO), which simultaneously finds the scan at a similar place and estimates their relative orientation. The orientation can further be used as the initial value for the down-stream local optimal metric pose estimation, improving the pose estimation especially when a large orientation between the current scan and retrieved scan exists. Our key idea is to transform the feature into the frequency domain. We utilize the magnitude of the spectrum as the place signature, which is theoretically rotation-invariant. In addition, based on the differentiable phase correlation, we can efficiently estimate the global optimal relative orientation using the spectrum. With such structural constraints, the network can be learned in an end-to-end manner, and the backbone is fully shared by the two tasks, achieving interpretability and light weight. Finally, DiSCO is validated on three datasets with long-term outdoor conditions, showing better performance than the compared methods.