Gravitational flexion, caused by derivatives of the gravitational tidal field, is potentially important for the analysis of the dark matter distribution in gravitational lenses, such as galaxy clusters or the dark matter haloes of galaxies. Flexion estimates rely on measurements of galaxy-shape distortions with spin-1 and spin-3 symmetry. We show in this paper that and how such distortions are generally caused not only by the flexion itself, but also by coupling terms of the form (shear × flexion), which have hitherto been neglected. Similar coupling terms occur between intrinsic galaxy ellipticities and the flexion. We show, by means of numerical tests, that neglecting these terms can introduce biases of up to 85 per cent on the F flexion and 150 per cent on the G flexion for galaxies with an intrinsic ellipticity dispersion of σε= 0.3. In general, this bias depends on the strength of the lensing fields, the ellipticity dispersion and the concentration of the lensed galaxies. We derive a new set of equations relating the measured spin-1 and spin-3 distortions to the lensing fields up to first order in the shear, the flexion, the product of shear and flexion, and the morphological properties of the galaxy sample. We show that this new description is accurate with a bias ≤ 7 per cent (spin-1 distortion) and ≤ 3 per cent (spin-3 distortion) even close to points where the flexion approach breaks down due to merging of multiple images. We propose an explanation why a spin-3 signal could not be measured yet and comment on the potential difficulties in using a model-fitting approach to measure the flexion signal.