The HyperCP Collaboration has recently reported the observation of three events for the decay Σ+→pμ+μ- with an invariant mass mμ+μ- for the muon-antimuon pair of ∼214 MeV. They suggest that a new particle state X may be needed to explain the observed mμ+μ- distribution. Motivated by this result, we study the properties of such a hypothetical particle. We first use K +→π+μ+μ- data to conclude that X cannot be a scalar or vector particle. We then collect existing constraints on a pseudoscalar or axial-vector X and find that these possibilities are still allowed as explanations for the HyperCP data. Finally we assume that the HyperCP data is indeed explained by a new pseudoscalar or axial-vector particle and use this to predict enhanced rates for K L→ππX→ππμ+μ- and Ω-→Ξ-X→Ξ-μ +μ-.