Light detection and ranging (LiDAR) point clouds and building information modeling (BIM) represent two distinct data modalities in the fields of robot perception and construction. These modalities originate from different sources and are associated with unique reference frames. The primary goal of this study is to align these modalities within a shared reference frame using a global registration approach, effectively enabling them to ``speak the same language''. To achieve this, we propose a cross-modality registration method, spanning from the front end to the back end. At the front end, we extract triangle descriptors by identifying walls and intersected corners, enabling the matching of corner triplets with a complexity independent of the BIM's size. For the back-end transformation estimation, we utilize the Hough transform to map the matched triplets to the transformation space and introduce a hierarchical voting mechanism to hypothesize multiple pose candidates. The final transformation is then verified using our designed occupancy-aware scoring method. To assess the effectiveness of our approach, we conducted real-world multi-session experiments in a large-scale university building, employing two different types of LiDAR sensors. We make the collected datasets and codes publicly available to benefit the community.