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, 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 operator in the framework of an effective field theory. The obtained bounds are C ∈ [−0.9, 1.4] and C ∈ [−0.8, 0.2], both at 95% confidence level. [Figure not available: see fulltext.].