Continuous function on the unit interval with commuting compositions

Question

Let $I$ be the unit interval and $f, g:I\to I$ be continuous functions. Assume that $f\circ g = g\circ f$.

This post shows that if $f$ and $g$ in addition are assumed to be increasing then $f$ and $g$ have a common fixed point.

Question. Does the same conclusion hold without assuming that $f$ and $g$ are monotonically increasing?

I couldn't construct any examples where the same conclusion does not hold.

Some Observations.

1. If $\text{Fix}(f) = \{x\in I: f(x) = x\}$ and $\text{Fix}(g) = \{x\in I:\ g(x) = x\}$, then $\text{Fix}(g)$ is $f$-invariant and $\text{Fix}(f)$ is $g$-invariant.

2. The set $\{x\in I:\ f(x) = g(x)\}$ (which, as shown here, is not empty) is both $f$ and $g$-invariant.