Based on a sample of (1310.6 +/- 10.5) x 10(6) J/psi events collected with the BESIII detector operating at the BEPCII storage ring, a partial wave analysis of the decay J/psi -> gamma phi phi is performed in order to study the intermediate states. Results of the partial wave analysis show that the structures are predominantly 0(-+) states. The existence of the eta(2225) is confirmed, and its resonance parameters are measured. Two additional pseudoscalar states, the eta(2100) with a mass of 2050(-24-26)(+30+75) MeV/c(2) and a width of 250(-30-164)(+36+181) MeV/c(2) and the X(2500) with a mass of 2470(-19-23)(+15+101) MeV/c(2) and a width of 230(-35-33)(+64+56) MeV/c(2), are observed. In addition to these three pseudoscalar states, the scalar state f(0)(2100), and three tensor states, the f(2)(2010), f(2)(2300) and f(2)(2340), are observed in the process J/psi -> gamma phi phi The product branching fractions B(J/psi -> gamma X) x B(X -> phi phi) are reported.