Numerous business workflows involve printed forms, such as invoices or receipts, which are often manually digitalized to persistently search or store the data. As hardware scanners are costly and inflexible, smartphones are increasingly used for digitalization. Here, processing algorithms need to deal with prevailing environmental factors, such as shadows or crumples. Current state-of-the-art approaches learn supervised image dewarping models based on pairs of raw images and rectification meshes. The available results show promising predictive accuracies for dewarping, but generated errors still lead to sub-optimal information retrieval. In this paper, we explore the potential of improving dewarping models using additional, structured information in the form of invoice templates. We provide two core contributions: (1) A novel dataset, referred to as Inv3D, comprising synthetic and real-world high-resolution invoice images with structural templates, rectification meshes, and a multiplicity of per-pixel supervision signals. (2) A novel image dewarping algorithm, which extends the state-of-the-art approach GeoTr to leverage structural templates using attention. Our extensive evaluation includes an implementation of DewarpNet and shows that exploiting structured templates can improve the performance for image dewarping. We report superior performance for the proposed algorithm on our new benchmark for all metrics, including an improved local distortion of 26.1 %. We made our new dataset and all code publicly available at https://felixhertlein.github.io/inv3d.