The study of the spin-parity and tensor structure of the interactions of the recently discovered Higgs boson is performed using the H -> ZZ, Z gamma*, gamma*gamma* -> 4l, H -> WW -> l(nu)l(nu), and H -> gamma gamma decay modes. The full data set recorded by the CMS experiment during the LHC run 1 is used, corresponding to an integrated luminosity of up to 5.1 fb(-1) at a center-of-mass energy of 7 TeV and up to 19.7 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 gravitylike couplings, regardless of whether assumptions are made on the production mechanism. Any mixed-parity spin-one state is excluded in the ZZ and 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 ZZ, Z gamma, gamma gamma, and WW is investigated and limits on eleven anomalous contributions are set. Tighter constraints on anomalous HVV interactions are obtained by combining the HZZ and HWW measurements. All observations are consistent with the expectations for the standard model Higgs boson with the quantum numbers J(PC) = 0(++).