  Let $f$ and $g$ be two quasiregular maps in $\mathbb{R}^d$ that are of
transcendental type and also satisfy $f\circ g =g \circ f$. We show that if the
fast escaping sets of those functions are contained in their respective Julia
sets then those two functions must have the same Julia set. We also obtain the
same conclusion about commuting quasimeromorphic functions with infinite
backward orbit of infinity. Furthermore we show that permutable quasiregular
functions of the form $f$ and $g=\phi\circ f$, where $\phi$ is a quasiconformal
map, have the same Julia sets and that polynomial type quasiregular maps cannot
commute with transcendental type ones unless their degree is less than or equal
to their dilatation.
