I think the answer is here:
http://cellular.ci.ulsa.mx/comun/algebra/node65.html
Basically,
for some $PX = \lambda X$ and $QY = \mu Y$ the following holds:
$$(P \otimes Q)(X \otimes Y) = (\lambda\mu)(X \otimes Y)$$.
This I guess implies that $(X \otimes Y)$ are the eigenvectors of $(P...