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This Letter presents a combination of searches for Higgs boson pair (HH) production performed by the ATLAS and CMS Collaborations using proton-proton collision data sets recorded at $\sqrt{s} = 13$ TeV during the Large Hadron Collider Run 2, corresponding to integrated luminosities ranging between 126 and 140 $\mathrm{fb^{-1}}$. The upper limit at the 95% confidence level on the total HH signal strength, defined as the ratio of the measured cross section to the SM prediction, corresponds to 2.5, with an expected value of 1.7 (2.8) assuming the absence (presence) of the standard model (SM) HH signal. The strength of the HH signal is measured to be $0.8^{+0.9}_{-0.7}$ relative to the SM prediction. The observed significance is found to be 1.1 standard deviations whereas 1.3 are expected for the SM HH signal. Constraints are set on the Higgs boson trilinear self-coupling and on the couplings of two Higgs bosons to two vector bosons, both normalized to the SM predictions and denoted as $κ_λ$ and $κ_{2\mathrm{V}}$, respectively. The observed individual constraints at the 95% confidence level are $-0.71 < κ_λ< 6.1$ and $0.73 < κ_{2\mathrm{V}} < 1.3$, while the expected constraints assuming the presence of the SM HH signal are $-1.3 < κ_λ< 6.7$ and $0.66 < κ_{2\mathrm{V}} < 1.4$.