Studies of the spin, parity and tensor couplings of the Higgs boson in the H → ZZ∗ → 4ℓ, H → WW∗ → eνμν and H → γγ decay processes at the LHC are presented. The investigations are based on 25 fb−1 of pp collision data collected by the ATLAS experiment at √s = 7 TeV and √s = 8 TeV. The Standard Model (SM) Higgs boson hypothesis, corresponding to the quantum numbers JP = 0+, is tested against several alternative spin scenarios, including non-SM spin-0 and spin-2 models with universal and non-universal couplings to fermions and vector bosons. All tested alternative models are excluded in favour of the SM Higgs boson hypothesis at more than 99.9% confidence level. Using the H → ZZ∗ → 4ℓ and H → WW∗ → eνμν decays, the tensor structure of the interaction between the spin-0 boson and the SM vector bosons is also investigated. The observed distributions of variables sensitive to the non-SM tensor couplings are compatible with the SM predictions and constraints on the non-SM couplings are derived.