Consider the tent function T defined by $T(x)=\begin{cases} 2x & \text{if } 0\le x\le 1/2\\ 2(1-x) & \text{if } 1/2<x\le 1 \end{cases}$ and the functions $d(x) = 1-x$, $k(x) = \begin{cases} \frac{1}{2}-x & \text{if } 0\le x\le 1/2\\ \frac{3}{2}-x & \text{if } 1/2 < x \le 1. \end{cases}$
Then $d\circ T = T\circ k$.
I wonder if this is true in general, in the sense that:
For every measure-preserving map $T$ and invertible measure-preserving map $d$, does there always exist invertible measure-preserving map $k$ such that $d\circ T = T\circ k$ on $[0,1]$?