A study of the charge conjugation and parity ($CP$) properties of the interaction between the Higgs boson and top quarks is presented. Higgs bosons are identified via the diphoton decay channel ($H\to \gamma \gamma$), and their production in association with a top quark pair ($t\overline{t}H$) or single top quark ($tH$) is studied. The analysis uses $139{\mathrm{fb}}^{-1}$ of proton–proton collision data recorded at a center-of-mass energy of $\sqrt{s}=13\mathrm{TeV}$ with the ATLAS detector at the Large Hadron Collider. Assuming a $CP$-even coupling, the $t\overline{t}H$ process is observed with a significance of 5.2 standard deviations. The measured cross section times $H\to \gamma \gamma$ branching ratio is ${1.64}_{-0.36}^{+0.38}(\text{stat}{)}_{-0.14}^{+0.17}\left(\text{sys}\right)\mathrm{fb}$, and the measured rate for $t\overline{t}H$ is ${1.43}_{-0.31}^{+0.33}(\text{stat}{)}_{-0.15}^{+0.21}\left(\text{sys}\right)$ times the Standard Model expectation. The $tH$ production process is not observed and an upper limit on its rate of 12 times the Standard Model expectation is set. A $CP$-mixing angle greater (less) than 43 $(-43)°$ is excluded at 95% confidence level.