In this note, we use techniques from integrable systems to study relations between gauge theories. The Gauge/Bethe correspondence, introduced by Nekrasov and Shatashvili, identifies the supersymmetric ground states of an N=(2,2) supersymmetric gauge theory in two dimensions with the Bethe states of a quantum integrable system. We make use of this correspondence to relate three different quiver gauge theories which correspond to three different formulations of the Bethe equations of an integrable spin chain called the tJ model.