Spinning LiDAR data are prevalent for 3D vision tasks. Since LiDAR data is presented in the form of point clouds, expensive 3D operations are usually required. This paper revisits spinning LiDAR scan formation and presents a cylindrical range image representation with a ray-wise projection/unprojection model. It is built upon raw scans and supports lossless conversion from 2D to 3D, allowing fast 2D operations, including 2D index-based neighbor search and downsampling. We then propose, to the best of our knowledge, the first multi-scale registration and dense signed distance function (SDF) reconstruction system for LiDAR range images. We further collect a dataset of indoor and outdoor LiDAR scenes in the posed range image format. A comprehensive evaluation of registration and reconstruction is conducted on the proposed dataset and the KITTI dataset. Experiments demonstrate that our approach outperforms surface reconstruction baselines and achieves similar performance to state-of-the-art LiDAR registration methods, including a modern learning-based registration approach. Thanks to the simplicity, our registration runs at 100Hz and SDF reconstruction in real time. The dataset and a modularized C++/Python toolbox will be released.