LiDAR relocalization plays a crucial role in many fields, including robotics, autonomous driving, and computer vision. LiDAR-based retrieval from a database typically incurs high computation storage costs and can lead to globally inaccurate pose estimations if the database is too sparse. On the other hand, pose regression methods take images or point clouds as inputs and directly regress global poses in an end-to-end manner. They do not perform database matching and are more computationally efficient than retrieval techniques. We propose HypLiLoc, a new model for LiDAR pose regression. We use two branched backbones to extract 3D features and 2D projection features, respectively. We consider multi-modal feature fusion in both Euclidean and hyperbolic spaces to obtain more effective feature representations. Experimental results indicate that HypLiLoc achieves state-of-the-art performance in both outdoor and indoor datasets. We also conduct extensive ablation studies on the framework design, which demonstrate the effectiveness of multi-modal feature extraction and multi-space embedding. Our code is released at: https://github.com/sijieaaa/HypLiLoc