T-cell receptors (TCR) play an important role in the adaptive immune system as they recognize pathogen- or cancer-based epitopes and thus initiate the cell-mediated immune response. Therefore there exists a growing interest in the optimization of TCRs for medical purposes like adoptive T-cell therapy. However, the molecular mechanisms behind T-cell signaling are still predominantly unknown. For small sets of TCRs it was observed that the angle between their Vα- and Vβ-domains, which bind the epitope, can vary and might be important for epitope recognition. Here we present a comprehensive, quantitative study of the variation in the Vα/Vβ interdomain-angle and its influence on epitope recognition, performing a systematic bioinformatics analysis based on a representative set of experimental TCR structures. For this purpose we developed a new, cuboid-based superpositioning method, which allows a unique, quantitative analysis of the Vα/Vβ-angles. Angle-based clustering led to six significantly different clusters. Analysis of these clusters revealed the unexpected result that the angle is predominantly influenced by the TCR-clonotype, whereas the bound epitope has only a minor influence. Furthermore we could identify a previously unknown center of rotation (CoR), which is shared by all TCRs. All TCR geometries can be obtained by rotation around this center, rendering it a new, common TCR feature with the potential of improving the accuracy of TCR structure prediction considerably. The importance of Vα/Vβ rotation for signaling was confirmed as we observed larger variances in the Vα/Vβ-angles in unbound TCRs compared to epitope-bound TCRs. Our results strongly support a two-step mechanism for TCR-epitope: First, preformation of a flexible TCR geometry in the unbound state and second, locking of the Vα/Vβ-angle in a TCR-type specific geometry upon epitope-MHC association, the latter being driven by rotation around the unique center of rotation.