Category-level object pose estimation aims to predict the 6D pose and 3D size of objects within given categories. Existing approaches for this task rely solely on 6D poses as supervisory signals without explicitly capturing the intrinsic continuity of poses, leading to inconsistencies in predictions and reduced generalization to unseen poses. To address this limitation, we propose HRC-Pose, a novel depth-only framework for category-level object pose estimation, which leverages contrastive learning to learn point cloud representations that preserve the continuity of 6D poses. HRC-Pose decouples object pose into rotation and translation components, which are separately encoded and leveraged throughout the network. Specifically, we introduce a contrastive learning strategy for multi-task, multi-category scenarios based on our 6D pose-aware hierarchical ranking scheme, which contrasts point clouds from multiple categories by considering rotational and translational differences as well as categorical information. We further design pose estimation modules that separately process the learned rotation-aware and translation-aware embeddings. Our experiments demonstrate that HRC-Pose successfully learns continuous feature spaces. Results on REAL275 and CAMERA25 benchmarks show that our method consistently outperforms existing depth-only state-of-the-art methods and runs in real-time, demonstrating its effectiveness and potential for real-world applications. Our code is at https://github.com/zhujunli1993/HRC-Pose.