A simultaneous measurement of the three components of the top-quark and top-antiquark polarisation vectors in t-channel single-top-quark production is presented. This analysis is based on data from proton–proton collisions at a centre-of-mass energy of 13 TeV corresponding to an integrated luminosity of 139 fb⁻¹, collected with the ATLAS detector at the LHC. Selected events contain exactly one isolated electron or muon, large missing transverse momentum and exactly two jets, one being b-tagged. Stringent selection requirements are applied to discriminate t-channel single-top-quark events from the background contributions. The top-quark and top-antiquark polarisation vectors are measured from the distributions of the direction cosines of the charged-lepton momentum in the top-quark rest frame. The three components of the polarisation vector for the selected top-quark event sample are Px′ = 0.01 ± 0.18, Py′ = −0.029 ± 0.027, Pz′ = 0.91 ± 0.10 and for the top-antiquark event sample they are Px′ = −0.02 ± 0.20, Py′ = −0.007 ± 0.051, Pz′ = 0.79 ± 0.16. Normalised differential cross-sections corrected to a fiducial region at the stable-particle level are presented as a function of the charged-lepton angles for top-quark and top-antiquark events inclusively and separately. These measurements are in agreement with Standard Model predictions. The angular differential cross-sections are used to derive bounds on the complex Wilson coefficient of the dimension-six O_tW operator in the framework of an effective field theory. The obtained bounds are C_tW ∈ [−0.9, 1.4] and C_itW ∈ [−0.8, 0.2], both at 95% confidence level. [Figure not available: see fulltext.].