  A study of the charge conjugation and parity ($CP$) properties of the
interaction between the Higgs boson and $\tau$-leptons is presented. The study
is based on a measurement of $CP$-sensitive angular observables defined by the
visible decay products of $\tau$-lepton decays, where at least one hadronic
decay is required. 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 $CP$-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 assuming the Standard Model hypothesis for the
$H\rightarrow\tau\tau$ signal strength, 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 $CP$-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.
