The processes γγ → ωϕ, ϕϕ, and ωω are measured using an 870 fb(-1) data sample collected with the Belle detector at the KEKB asymmetric-energy e+ e- collider. Production of vector meson pairs is clearly observed and their cross sections are measured for masses that range from threshold to 4.0 GeV. In addition to signals from well established spin-zero and spin-two charmonium states, there are resonant structures below charmonium threshold, which have not been previously observed. We report a spin-parity analysis for the new structures and determine the products of the η(c), χ(c0), and χ(c2) two-photon decay widths and branching fractions to ωϕ, ϕϕ, and ωω.