  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.
