There is evidence of a second neutral pion from: (1) the anomalous branching ratios in the reactions p-bar p -> pi pi and p-bar d -> pi pi N, and (2) the 1960's results of Tsai-Chu et al. for antinucleon annihilation stars in emulsions. The anomaly of (1) is eliminated if the two neutral pions in the reactions p-bar p -> pi_0 pi_0 and p-bar d -> pi_0 pi_0 n are not identical. Tsai-Chu et al. observed a second neutral pion that ``decays more rapidly into electron pairs with larger opening angles and more frequently into double pairs.'' One antineutron annihilation event produced three neutral particles (each with a mass of 135 +/- 14 MeV), and each decayed into four electrons with much wider opening angles than those of the internal conversion electrons seen in pi-zero decays. The larger opening angles and much more frequent double pair production could be caused by neutral pions with a lifetime so short that they sometimes decay into photon pairs before they can leave the annihilation nucleus (e.g., Ag) of the emulsion. We discuss several methods of searching for this second neutral pion.