ABSTRACT The distribution of mass in galaxy-scale strong gravitational lenses is often modelled as an elliptical power-law plus ‘external shear’, which notionally accounts for neighbouring galaxies and cosmic shear along our line of sight. A small amount of external shear could come from these sources, but we show that the vast majority does not. Except in a handful of rare systems, the best-fitting values do not correlate with independent measurements of line-of-sight shear: from weak lensing in 45 Hubble Space Telescope images, or in 50 mock images of lenses with complex distributions of mass. Instead, the best-fit external shear is aligned with the major or minor axis of 88 per cent of lens galaxies; and the amplitude of the external shear increases if that galaxy is discy. We conclude that ‘external shear’ attached to a power-law model is not physically meaningful, but a fudge to compensate for lack of model complexity. Since it biases other model parameters that are interpreted as physically meaningful in several science analyses (e.g. measuring galaxy evolution, dark matter physics or cosmological parameters), we recommend that future studies of galaxy-scale strong lensing should employ more flexible mass models.