A robust, yet computationally efficient signature for describing 3D shape remains a challenge for 3D computer vision and related applications. Having a signature that is generalizable across a wider range of datasets becomes another important research issue. This paper proposes a novel Hybrid signature, the Augmented Point Pair Signature (HAPPS), that is robust, highly discriminating, efficient, and capable of effectively representing 3D point cloud and polygon mesh surfaces. We tested the overall performances of HAPPS on three standardized benchmark datasets for 3D shape retrieval: The Shape Retrieval Contest 2018 (SHREC'18) protein shapes benchmark, with 2,267 protein conformers, SHREC'17 Point cloud Retrieval of Nonrigid Toys (PRoNTo), with 100 3D point clouds, and SHREC'10 Nonrigid shape retrieval having 200 triangular meshes. Using 6 standard retrieval performance metrics to evaluate our results, we demonstrated the superiority of our HAPPS retrieval method over several other state-of-the-art methods for the SHREC'18 protein dataset, while also competing side-by-side with the best 2 performing methods for the other benchmark datasets.