We study the processes e + e − → K 0 S K 0 L γ , K 0 S K 0 L π + π − γ , K 0 S K 0 S π + π − γ , and K 0 S K 0 S K + K − γ , where the photon is radiated from the initial state, providing cross section measurements for the hadronic states over a continuum of center-of-mass energies. The results are based on 469   fb − 1 of data collected with the BABAR detector at SLAC. We observe the ϕ ( 1020 ) resonance in the K 0 S K 0 L final state and measure the product of its electronic width and branching fraction with about 3% uncertainty. We present a measurement of the e + e − → K 0 S K 0 L cross section in the energy range from 1.06 to 2.2 GeV and observe the production of a resonance at 1.67 GeV. We present the first measurements of the e + e − → K 0 S K 0 L π + π − , K 0 S K 0 S π + π − , and K 0 S K 0 S K + K − cross sections and study the intermediate resonance structures. We obtain the first observations of J / ψ decay to the K 0 S K 0 L π + π − , K 0 S K 0 S π + π − , and K 0 S K 0 S K + K − final states.