In response to a question posed by O.E. Lanford III, it is shown that for each \(\mu>0\) there is a differentiable and non-linearizable interval map $g$ with non-vanishing derivative defined on a neighborhood of a fixed point $0$ with \(g^\prime(0)=\mu\) such that $g$ has infinitely many differentiable and non-linearizable orientation-reversing composition square roots with non-vanishing first derivatives on a neighborhood of $0$.
"Non-Uniqueness of Composition Square Roots." Real Anal. Exchange 26 (2) 861 - 866, 2000/2001.