Two entire functions \(f\) and \(g\) are called permutable if \(f\circ g=g\circ f\). It is an open question whether permutable entire functions must have the same Julia set. (For rational functions this is a classical result of Fatou and Julia.) \textit{I. N. Baker} [Proc. Lond. Math. Soc., III. Ser. 49, 563-576 (1984; Zbl 0523.30017)] showed that this is the case if \(f=g+b\) for some constant \(b\). This was generalized by \textit{K. K. Poon} and \textit{C.-C. Yang} [Ann. Pol. Math. 68, 159-163 (1998; Zbl 0896.30021)] who showed that the Julia sets of permutable entire functions \(f\) and \(g\) are equal if \(f=ag+b\) for constants \(a,b\). Here this result is further generalized. It is shown that it suffices to assume that \(q(f)=aq(g)+b\) for constants \(a,b\) and a non-constant polynomial \(q\). Analogous results are obtained not only for the Julia sets, but also for sets where the iterates tend to \(\infty\) with a certain speed. Further results address the question when the Fatou components of \(f\) and \(g\) are of the same type according to the classification of Fatou components into wandering domains and (preimages of) Böttcher domains, Schröder domains, Landau domains, Siegel discs and Baker domains.