Image-to-point cloud registration aims to determine the relative camera pose of an RGB image with respect to a point cloud. It plays an important role in camera localization within pre-built LiDAR maps. Despite the modality gaps, most learning-based methods establish 2D-3D point correspondences in feature space without any feedback mechanism for iterative optimization, resulting in poor accuracy and interpretability. In this paper, we propose to reformulate the registration procedure as an iterative Markov decision process, allowing for incremental adjustments to the camera pose based on each intermediate state. To achieve this, we employ reinforcement learning to develop a cross-modal registration agent (CMR-Agent), and use imitation learning to initialize its registration policy for stability and quick-start of the training. According to the cross-modal observations, we propose a 2D-3D hybrid state representation that fully exploits the fine-grained features of RGB images while reducing the useless neutral states caused by the spatial truncation of camera frustum. Additionally, the overall framework is well-designed to efficiently reuse one-shot cross-modal embeddings, avoiding repetitive and time-consuming feature extraction. Extensive experiments on the KITTI-Odometry and NuScenes datasets demonstrate that CMR-Agent achieves competitive accuracy and efficiency in registration. Once the one-shot embeddings are completed, each iteration only takes a few milliseconds.