At the moment I am reading the book "An Introduction to Chaotic Dynamical Systems" by Robert L. Devaney.
In Chapter 18, "Kneading Theory", he introduces following stuff.
Definition Let $f:[0,1] \rightarrow [0,1]$ be a $C^1([0,1])$ map. $f$ is unimodal if $f(0) = f(1) = 0$ and $f$ has a unique critical point $c \in (0,1)$.
Now let $f$ be a fixed unimodal map.
Definition Let $x \in [0,1]$. The itinerary of $x$ under $f$ is the infinite sequence $S(x)= (s_0,s_1,s_2,\dots)$ where $$s_j = \begin{cases} 0 \quad \text{if} \quad f^j(x) < c\\ 1 \quad \text{if} \quad f^j(x) > c\\ C \quad \text{if} \quad f^j(x) = c. \end{cases}$$ where $C$ is just some symbol.
We first define an order on $\{0,C,1\}$ by $0 < C$, $C < 1$ and $0 < 1$.
Then he defines an ordering on the space $\Sigma_C := \{(x_n)_{n \in \mathbb{N}}| x_n \in \{0,C,1\} \quad \forall n \in \mathbb{N}\}$.
Definition We say $s,t \in \Sigma_C$ have discrepancy $n$ if $s_0 = t_0$, $s_1 = t_1$, $\dots$, $s_{n-1} = t_{n-1}$ and $s_n \neq t_n$. Additionally $\tau_n(s) = \#\{s_i | i \in \{0,\dots, n\} \text{ and } s_i = 1 \}$.
Definition Let $s,t \in \Sigma_C$ and assume $s$ and $t$ have discrepancy $n$. We say $s \prec t$ if either ($\tau_{n-1}(s)$ is even and $s_n<t_n$) or ($\tau_{n-1}(s)$ is odd and $s_n>t_n$).
This ordering hat the property that for $x,y \in [0,1]$ holds
if $S(x) \prec S(y)$, the $x<y$.
if $x<y$, then $S(x) \preccurlyeq S(y)$.
But is there a deeper intuition behind, more than just this technical properties?