  Massive elliptical galaxies can display structures that deviate from a pure
elliptical shape, such as a twist of the principal axis or variations in the
axis ratio with galactocentric distance. Although satisfactory lens modeling is
generally achieved without accounting for these azimuthal structures, the
question about their impact on inferred lens parameters remains, in particular,
on time delays as they are used in time-delay cosmography. This paper aims at
characterizing these effects and quantifying their impact considering realistic
amplitudes of the variations. We achieved this goal by creating mock lensing
galaxies with morphologies based on two data sets: observational data of local
elliptical galaxies, and hydrodynamical simulations of elliptical galaxies at a
typical lens redshift. We then simulated images of the lensing systems with
space-based data quality and modeled them in a standard way to assess the
impact of a lack of azimuthal freedom in the lens model. We find that twists in
lensing galaxies are easily absorbed in homoeidal lens models by a change in
orientation of the lens up to 10{\deg} with respect to the reference
orientation at the Einstein radius, and of the shear by up to 20{\deg} with
respect to the input shear orientation. The ellipticity gradients, on the other
hand, can introduce a substantial amount of shear that may impact the radial
mass model and consequently bias $H_0$, up to 10 km/s/Mpc. However, we find
that light is a good tracer of azimuthal structures, meaning that direct
imaging should be capable of diagnosing their presence. This in turn implies
that such a large bias is unlikely to be unaccounted for in standard modeling
practices. Furthermore, the overall impact of twists and ellipticity gradients
averages out at a population level. For the galaxy populations we considered,
the cosmological inference remains unbiased.
