Arc length is invariant under every continuous surjective monotone reparametrization, including pauses and reversal

Suppose first that $\phi$ is nondecreasing. The image under $\phi$ of any partition of $[c,d]$ is a nondecreasing finite list from $a$ to $b$; deleting repetitions produces a partition of $[a,b]$ with the same polygonal sum for $\gamma$.

Conversely, for a partition $a=t_0<\cdots <t_m=b$, choose one $s_i\in {\phi }^{-1}(t_i)$ for each of its finitely many values. Monotonicity forces $s_0<\cdots <s_m$, after taking $s_0=c$ and $s_m=d$, and the resulting polygonal sum of $\gamma \circ \phi$ equals that of $\gamma$.

Hence every polygonal sum of $\gamma \circ \phi$ is at most $L(\gamma )$, so $L(\gamma \circ \phi )\le L(\gamma )$.

Taking the supremum over target partitions gives $L(\gamma )\le L(\gamma \circ \phi )$, proving equality in the nondecreasing case.

If $\phi$ is nonincreasing, reverse the order of every finite list in steps 1.1 and 1.2; Euclidean chord lengths are symmetric, so the same two inequalities hold.

If $[c,d]$ is a singleton, so is its image $[a,b]$, and the singleton convention in [L2] gives both lengths as zero. If instead $a=b$ while $c<d$, then $\gamma \circ \phi$ is constant, so every polygonal sum for it vanishes and $L_{[c,d]}(\gamma \circ \phi )=0$, while $L_{[a,b]}(\gamma )=0$ by the same convention.