A bstract 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 $$ {P}_{x^{\prime }} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:msub> </mml:math> = 0 . 01 ± 0 . 18, $$ {P}_{y^{\prime }} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:msup> <mml:mi>y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:msub> </mml:math> = − 0 . 029 ± 0 . 027, $$ {P}_{z^{\prime }} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:msup> <mml:mi>z</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:msub> </mml:math> = 0 . 91 ± 0 . 10 and for the top-antiquark event sample they are $$ {P}_{x^{\prime }} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:msub> </mml:math> = − 0 . 02 ± 0 . 20, $$ {P}_{y^{\prime }} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:msup> <mml:mi>y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:msub> </mml:math> = − 0 . 007 ± 0 . 051, $$ {P}_{z^{\prime }} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:msup> <mml:mi>z</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:msub> </mml:math> = 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 $$ \mathcal{O} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> </mml:math> 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.