Based on (10.09 +/- 0.04) x 10(9) J/psi events collected with the BESIII detector operating at the BEPCII collider, a partial wave analysis of the decay J/psi -> phi pi(0 eta). is performed. We observe for the first time two new structures on the phi eta invariant mass distribution, with significances exceeding 27 sigma and 13 sigma; the first with J(PC) = 1(+-), mass M = (1908 +/- 6(stat)(-4)(+8) (sys)) MeV/c(2), and width Gamma = (175 +/- 13(stat)(-16)(+7) (sys)) MeV, the second with J(PC) = 1(--), mass M = (1992 +/- 12 (stat)(-6)(+15) (sys)) MeV/c(2), and width Gamma = (132 +/- 22(stat)(-4)(+17) (sys)) MeV. These measurements provide important input for the strangeonium spectrum. In addition, the f(0) (980) - a(0)(980)(0) mixing signal in J/psi -> phi f(0)(980) -> phi a(0)(980)(0) and the corresponding electromagnetic decay J/psi -> phi a(0)(980)(0) are measured with improved precision, providing crucial information to understand the nature of a(0)(980)(0) and f(0)(980).