Measurements of joint-polarisation states of W and Z gauge bosons in W^±Z production are presented. The data set used corresponds to an integrated luminosity of 139 fb⁻¹ of proton–proton collisions at a centre-of-mass energy of 13 TeV recorded by the ATLAS detector at the CERN Large Hadron Collider. The W^±Z candidate events are reconstructed using leptonic decay modes of the gauge bosons into electrons and muons. The simultaneous pair-production of longitudinally polarised vector bosons is measured for the first time with a significance of 7.1 standard deviations. The measured joint helicity fractions integrated over the fiducial region are f₀₀=0.067±0.010, f_0T=0.110±0.029, f_T0=0.179±0.023 and f_TT=0.644±0.032, in agreement with the next-to-leading-order Standard Model predictions. Individual helicity fractions of the W and Z bosons are also measured and found to be consistent with joint helicity fractions within the expected amounts of correlation. Both the joint and individual helicity fractions are also measured separately in W⁺Z and W⁻Z events. Inclusive and differential cross sections for several kinematic observables sensitive to polarisation are presented.