αβ T cell receptors (TCRs) recognize peptides presented by major histocompatibility complex (MHC) proteins using multiple complementarity-determining region (CDR) loops. TCRs display an array of poorly understood recognition properties, including specificity, crossreactivity and MHC restriction. Here we report a comprehensive thermodynamic deconstruction of the interaction between the A6 TCR and the Tax peptide presented by the class I MHC HLA-A*0201, uncovering the physical basis for the receptor's recognition properties. Broadly, our findings are in conflict with widely held generalities regarding TCR recognition, such as the relative contributions of central and peripheral peptide residues and the roles of the hypervariable and germline CDR loops in engaging peptide and MHC. Instead, we find that the recognition properties of the receptor emerge from the need to engage the composite peptide/MHC surface, with the receptor utilizing its CDR loops in a cooperative fashion such that specificity, crossreactivity and MHC restriction are inextricably linked.