A study of the charge conjugation and parity (textit{CP}) properties of the interaction between the Higgs boson and tau -leptons is presented. The study is based on a measurement of textit{CP}-sensitive angular observables defined by the visible decay products of tau -leptons produced in Higgs boson decays. The analysis uses 139 fb^{-1} of proton–proton collision data recorded at a centre-of-mass energy of sqrt{s}= 13 TeV with the ATLAS detector at the Large Hadron Collider. Contributions from textit{CP}-violating interactions between the Higgs boson and tau -leptons are described by a single mixing angle parameter phi _{tau } in the generalised Yukawa interaction. Without constraining the Hrightarrow tau tau signal strength to its expected value under the Standard Model hypothesis, the mixing angle phi _{tau } is measured to be 9^{circ } pm 16^{circ }, with an expected value of 0^{circ } pm 28^{circ } at the 68% confidence level. The pure textit{CP}-odd hypothesis is disfavoured at a level of 3.4 standard deviations. The results are compatible with the predictions for the Higgs boson in the Standard Model.