We report new, extremely precise, photopolarimetry of the rapidly-rotating A0 main-sequence star ζ Aql, covering the wavelength range ∼400-900nm, which reveals a rotationally-induced signal. We model the polarimetry, together with the flux distribution and line profiles, in the framework of Roche geometry with ω-model gravity darkening, to establish the stellar parameters. An additional constraint is provided by TESS photometry, which shows variability with a period, P phot , of 11.1 hr. Modelling based on solid-body surface rotation gives rotation periods, P rot , that are in only marginal agreement with this value. We compute new ester stellar-structure models to predict horizontal surface velocity fields, which depart from solid-body rotation at only the ∼2% level (consistent with a reasonably strong empirical upper limit on differential rotation derived from the line-profile analysis). These models bring the equatorial rotation period, P rot(e) , into agreement with P phot , without requiring any 'fine tuning' (for the Gaia parallax). We confirm that surface abundances are significantly subsolar ([M/H] −0.5). The star's basic parameters are established with reasonably good precision:• , and ω e /ω c = 0.95 ± 0.02. Comparison with single-star, solar-abundance stellar-evolution models incorporating rotational effects shows excellent agreement (but somewhat poorer agreement for models at [M/H] −0.4).