Let \(M_n\) denote the space of complex \(n\times n\) matrices, and let~\(A,B,C\in M_n\). The \(C\)-numerical range of~\(A\) is defined by \(W_C(A)=\{\mathrm{tr}\,(CU^*AU):\,U\in M_n\, \mathrm{is\,unitary}\}\). The authors study the equation (1)~\(W_C(AB)=W_C(BA)\). First, they prove this if \(A\) and \(B\) are symmetric and \(C\) is normal. Second, they discuss the cases \(n=2\) and \(n=3\). Finally, they show that (1) holds if \(C\) is normal and \(A\) and \(B\) are Toeplitz matrices or matrices symmetric with respect to their backward diagonals or continuant matrices (i.e., tridiagonal matrices with entries \(1\) in the first diagonal above the main diagonal and \(-1\) in the first diagonal below it).