The large amount of data collected by LiDAR sensors brings the issue of LiDAR point cloud compression (PCC). Previous works on LiDAR PCC have used range image representations and followed the predictive coding paradigm to create a basic prototype of a coding framework. However, their prediction methods give an inaccurate result due to the negligence of invalid pixels in range images and the omission of future frames in the time step. Moreover, their handcrafted design of residual coding methods could not fully exploit spatial redundancy. To remedy this, we propose a coding framework BIRD-PCC. Our prediction module is aware of the coordinates of invalid pixels in range images and takes a bidirectional scheme. Also, we introduce a deep-learned residual coding module that can further exploit spatial redundancy within a residual frame. Experiments conducted on SemanticKITTI and KITTI-360 datasets show that BIRD-PCC outperforms other methods in most bitrate conditions and generalizes well to unseen environments.