We investigate how the CP quantum numbers of a neutral Higgs boson or spin-zero resonance Φ, produced at the CERN Large Hadron Collider, can be determined in its τ-pair decay mode Φ → τ − τ + . We use a method [1] based on the distributions of two angles and apply it to the major 1-prong τ decays. We show for the resulting dilepton, lepton-pion, and two-pion final states that appropriate selection cuts significantly enhance the discriminating power of these observables. From our analysis we conclude that, provided a Higgs boson will be found at the LHC, it appears feasible to collect the event numbers needed to discriminate between a CP-even and CP-odd Higgs boson and/or between Higgs boson(s) with CP-conserving and CP-violating couplings after several years of high-luminosity runs.