Instance-level object re-identification is a fundamental computer vision task, with applications from image retrieval to intelligent monitoring and fraud detection. In this work, we propose the novel task of damaged object re-identification, which aims at distinguishing changes in visual appearance due to deformations or missing parts from subtle intra-class variations. To explore this task, we leverage the power of computer-generated imagery to create, in a semi-automatic fashion, high-quality synthetic images of the same bike before and after a damage occurs. The resulting dataset, Bent & Broken Bicycles (BBBicycles), contains 39,200 images and 2,800 unique bike instances spanning 20 different bike models. As a baseline for this task, we propose TransReI3D, a multi-task, transformer-based deep network unifying damage detection (framed as a multi-label classification task) with object re-identification. The BBBicycles dataset is available at https://huggingface.co/datasets/GrainsPolito/BBBicycles