  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 $\times$ 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% on the $F$
flexion and 150% on the $G$ flexion for galaxies with an intrinsic ellipticity
dispersion of $\sigma_{\epsilon}=0.3$. 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 $\leq 7%$ (spin-1 distortion) and $\leq 3%$
(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 difficulties in using a
model-fitting approach to measure the flexion signal.
