The variable domains of antibodies and T-Cell receptors (TCRs) share similar structures. Both molecules act as sensors for the immune system but recognise their respective antigens in different ways. Antibodies bind to a diverse set of antigenic shapes whilst TCRs only recognise linear peptides presented by a major histocompatibility complex (MHC). The antigen specificity and affinity of both receptors is determined primarily by the sequence and structure of their complementarity determining regions (CDRs). In antibodies the binding site is also known to be affected by the relative orientation of the variable domains, VH and VL. Here, the corresponding property for TCRs, the Vβ-Vα orientation, is investigated and compared with that of antibodies. We find that TCR and antibody orientations are distinct. General antibody orientations are found to be incompatible with binding to the MHC in a canonical TCR-like mode. Finally, factors that cause the orientation of TCRs and antibodies to be different are investigated. Packing of the long Vα CDR3 in the domain-domain interface is found to be influential. In antibodies, a similar packing affect can be achieved using a bulky residue at IMGT position 50 on the VH domain. Along with IMGT VH 50, other positions are identified that may help to promote a TCR-like orientation in antibodies. These positions should provide useful considerations in the engineering of therapeutic TCR-like antibodies.