This paper shows that chiral carbon nanotubes (CNTs) under twisting show highly dependent strength and fracture toughness on (i) the level of axial tension and (ii) the direction of twisting. To achieve this conclusion, numerous computational intensive molecular dynamics simulations of chiral (6,3) CNTs were performed using the LAMMPS code and the AIREBO potential for C–C bonds. First, we have studied the influence of the tension level and the direction of twisting on the buckling and failure of the chiral CNTs. We show that the applied torques and angles of twist at the buckling and failure stages strongly depend on the level of tensile loading and on the twist direction. In order to explain such noticeable anisotropic behaviour, we have studied the evolution of two kinematic variables (C–C bond length and hexagonal cell angle) as a function of the twist–tension rate. We conclude that the anisotropic failure of chiral CNTs strongly depends on the type of rupture mechanism, which varies with (i) the level of axial tension and (ii) the direction of twisting. Finally, we summarize the information by means of tension–twisting interaction diagrams, which are very dissimilar for direct and inverse twisting. We conclude that inverse twisting influences very negatively the collapse behaviour of chiral CNTs under tension. On the other hand, for low-to-moderate twist–tension rates, the coupling between tension and direct twisting is highly beneficial for the behaviour of chiral CNTs against collapse. This scenario changes dramatically for moderate-to-high twist–tension rates, as the coupling between tension and direct twisting becomes highly detrimental for the chiral CNT collapse process. These original findings might be relevant for the design of nano-devices made of chiral CNTs, as we show that the fracture toughness of chiral CNTs under tension can be highly increased if low-to-moderate direct twisting is added.