A remarkable symmetry has been observed between the diagonal, nonlinear, current-voltage (I-V-xx) characteristics taken in the fractional quantum Hall effect (FQHE) liquid state of the two-dimensional electron system and those taken in the bordering insulating phase. When properly selected, the I-V-xx traces in the FQHE regime are identical, within experimental errors, to V-xx-I traces in the insulator, that is, with the roles of the currents and voltages exchanged. These results can be interpreted as evidence for the existence of charge-flux duality symmetry in the system: