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The correct ego-motion estimation basically relies on the understanding of correspondences between adjacent LiDAR scans. However, given the complex scenarios and the low-resolution LiDAR, finding reliable structures for identifying correspondences can be challenging. In this paper, we delve into structure reliability for accurate self-supervised ego-motion estimation and aim to alleviate the influence of unreliable structures in training, inference and mapping phases. We improve the self-supervised LiDAR odometry substantially from three aspects: 1) A two-stage odometry estimation network is developed, where we obtain the ego-motion by estimating a set of sub-region transformations and averaging them with a motion voting mechanism, to encourage the network focusing on representative structures. 2) The inherent alignment errors, which cannot be eliminated via ego-motion optimization, are down-weighted in losses based on the 3D point covariance estimations. 3) The discovered representative structures and learned point covariances are incorporated in the mapping module to improve the robustness of map construction. Our two-frame odometry outperforms the previous state of the arts by 16%/12% in terms of translational/rotational errors on the KITTI dataset and performs consistently well on the Apollo-Southbay datasets. We can even rival the fully supervised counterparts with our mapping module and more unlabeled training data.