A polynomial of the form \(f(x)-g(y),\) where x and y are disjoint finite sets of variables, is called a difference polynomial. Let \(P(x)-Q(y)\) and \(P^*(x)-Q^*(y)\) be two difference-polynomials having an irreducible common factor F. The main theorem of this article establishes the existence of a difference polynomial f(x)-g(y) which is divisible by F and has the property that \(P=\phi \circ f,Q=\phi \circ g,P^*=\phi^*\circ f\) and \(Q^*=\phi^*\circ g\) for some one- variable polynomials \(\phi\) and \(\phi^*\). This theorem is useful in studying the class of rational functions sharing a level curve in the complex plane.