Controlling charge transport through a single molecule connected to metallic electrodes remains one of the most fundamental challenges of nanoelectronics. Here we use electrochemical gating to reversibly tune the conductance of two different organic molecules, both containing anthraquinone (AQ) centers, over >1 order of magnitude. For electrode potentials outside the redox-active region, the effect of the gate is simply to shift the molecular energy levels relative to the metal Fermi level. At the redox potential, the conductance changes abruptly as the AQ unit is oxidized/reduced with an accompanying change in the conjugation pattern between linear and cross conjugation. The most significant change in conductance is observed when the electron pathway connecting the two electrodes is via the AQ unit. This is consistent with the expected occurrence of destructive quantum interference in that case. The experimental results are supported by an excellent agreement with ab initio transport calculations.