Abstract: The new particle X recently discovered by the ATLAS and CMS Collaborations in searches for the Higgs boson has been observed to decay into gamma gamma, ZZ* and WW*, but its spin and parity, J^P, remain a mystery, with J^P = 0^+ and 2^+ being open possibilities. We use PYTHIA and Delphes to simulate an analysis of the angular distribution of gg to X to gamma gamma decays in a full 2012 data set, including realistic background levels. We show that this angular distribution should provide strong discrimination between the possibilities of spin zero and spin two with graviton-like couplings: ~ 3 sigma if a conservative symmetric interpretation of the log-likelihood ratio (LLR) test statistic is used, and ~ 6 sigma if a less conservative asymmetric interpretation is used. The WW and ZZ couplings of the Standard Model Higgs boson and of a 2^+ particle with graviton-like couplings are both expected to exhibit custodial symmetry. We simulate the present ATLAS and CMS search strategies for X to WW* using PYTHIA and Delphes, and show that their efficiencies in the case of a spin-two particle with graviton-like couplings are a factor ~ 1.9 smaller than in the spin-zero case. On the other hand, the ratio of X_{2^+} to WW* and ZZ* branching ratios is larger than that in the 0^+ case by a factor ~ 1.3. We find that the current ATLAS and CMS results for X to WW* and X to ZZ* decays are compatible with custodial symmetry under both the spin-zero and -two hypotheses, and that the data expected to become available during 2012 are unlikely to discriminate significantly between these possibilities.