  It has been taken as granted that the observation of two independent
mass-squared differences necessarily fixes the number of underlying mass
eigenstates as three, and that the addition of a sterile neutrino provides an
additional mass-squared difference. The purpose of this Letter is to argue that
if one considers a sterile neutrino component that belongs to the (CP)^2 = - 1
sector, then both of the stated claims are false. We also outline how the
results reported here, when combined with the forthcoming MiniBooNE data and
other experiments, can help settle the issue of the CP properties of the
sterile neutrino; if such a component does indeed exist.
