The study of the spin-parity and tensor structure of the interactions of the recently discovered Higgs boson is performed using the $\mathrm{H} \rightarrow \mathrm{Z} \mathrm{Z}$, $\mathrm{Z}\gamma^*$, $\gamma^*\gamma^* \rightarrow 4\ell$, $\mathrm{H} \rightarrow \mathrm{W} \mathrm{W} \rightarrow \ell\nu\ell\nu$, and $\mathrm{H} \rightarrow \gamma\gamma$ decay modes. The full dataset recorded by the CMS experiment during the LHC Run 1 is used, corresponding to an integrated luminosity of up to 5.1~\mathrm{fb}^{-1} at a center-of-mass energy of 7 TeV and up to 19.7~\mathrm{fb}^{-1} at 8 TeV. A wide range of spin-two models is excluded at a 99% confidence level or higher, or at a 99.87% confidence level for the minimal gravity-like couplings, regardless of whether assumptions are made on the production mechanism. Any mixed-parity spin-one state is excluded in the $\mathrm{ZZ}$ and $\mathrm{WW }$ modes at a greater than 99.999% confidence level. Under the hypothesis that the resonance is a spin-zero boson, the tensor structure of the interactions of the Higgs boson with two vector bosons $\mathrm{Z} \mathrm{Z}$, $\mathrm{Z}\gamma$, $\gamma\gamma$, and $\mathrm{W} \mathrm{W}$ is investigated and limits on eleven anomalous contributions are set. Tighter constraints on anomalous $\mathrm{HVV}$ interactions are obtained by combining the $\mathrm{HZZ}$ and $\mathrm{HWW}$ measurements. All observations are consistent with the expectations for the standard model Higgs boson with the quantum numbers $J^{\mathrm{PC}}=0^{++}$.

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