  Maps $f,g\colon I\to I$ are called strongly commuting if $f\circ
g^{-1}=g^{-1}\circ f$. We show that strongly commuting, piecewise monotone maps
$f,g$ can be decomposed into a finite number of invariant intervals (or period
2 intervals) on which $f,g$ are either both open maps, or at least one of them
is monotone. As a consequence, we show that strongly commuting piecewise
monotone interval maps have a common fixed point. Results of the paper also
have implications in understanding dynamical properties of certain maps on
inverse limit spaces.
