To probe the W tb vertex structure, top-quark and W -boson polarisation observables are measured from t-channel single-top-quark events produced in proton-proton collisions at a centre-of-mass energy of 8 TeV. The dataset corresponds to an integrated luminosity of 20.2 fb−1, recorded with the ATLAS detector at the LHC. Selected events contain one isolated electron or muon, large missing transverse momentum and exactly two jets, with one of them identified as likely to contain a b-hadron. Stringent selection requirements are applied to discriminate t-channel single-top-quark events from background. The polarisation observables are extracted from asymmetries in angular distributions measured with respect to spin quantisation axes appropriately chosen for the top quark and the W boson. The asymmetry measurements are performed at parton level by correcting the observed angular distributions for detector effects and hadronisation after subtracting the background contributions. The measured top-quark and W -boson polarisation values are in agreement with the Standard Model predictions. Limits on the imaginary part of the anomalous coupling gR are also set from model-independent measurements.[Figure not available: see fulltext.].