Eigenvalues of Kronecker Product

I don't have Merris' book, but something seems to be forgotten in the question - in the present form it cannot be answered.

Consider, for instance, diagonal matrices $A$ and $B$ explicitly given by $$A=\operatorname{diag}\left\{\lambda_1,\lambda_2,\lambda_3\right\},\qquad B=\operatorname{diag}\left\{\omega_1,\omega_2\right\},$$ so that \begin{align} &A\otimes B+B\otimes A=\\ =\,&\operatorname{diag}\left\{2\lambda_1\omega_1,\lambda_2\omega_1+\lambda_1\omega_2, (\lambda_2+\lambda_3)\omega_1,(\lambda_1+\lambda_2)\omega_2,\lambda_3\omega_1+\lambda_2\omega_2,2\lambda_3\omega_2\right\}. \end{align} The spectrum of the last matrix is not invariant e.g. with respect to the exchange $\omega_1\leftrightarrow\omega_2$.

Furthermore, even setting $p=q$ as in the book referenced in the comment does not save the situation: again consider diagonal $2\times 2$ matrices $A,B$ and notice that the spectrum of $A\otimes B+ B\otimes A$, given by $\{2\lambda_1\omega_1,\lambda_1\omega_2+\lambda_2\omega_1,\lambda_1\omega_2+\lambda_2\omega_1,2\lambda_2\omega_2\}$ is not invariant w.r.t. the exchange of eigenvalues of one of them.