ABSTRACT We report new, extremely precise photopolarimetry of the rapidly-rotating A0 main-sequence star ζ Aql, covering the wavelength range ∼400–900 nm, 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, Pphot, of 11.1 h. Modelling based on solid-body surface rotation gives rotation periods, Prot, 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 per cent 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, Prot(e), into agreement with Pphot, 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: $M = 2.53\pm 0.16\, \mbox{M}_{\odot }$, log (L/L⊙) = 1.72± 0.02, $R_{\rm p}= 2.21\pm 0.02\, \mbox{R}_{\odot }$, Teff = 9693 ± 50 K, $i = 85{^{+5}_{-7}}^\circ$, and ωe/ωc = 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).