Reversal, concatenation, closed paths, and oriented piecewise-C1 reparametrizations

Let $\gamma :[a,b]\to {\mathbb{R}}^n$ be a piecewise-$C^1$ path in the sense of Paths in ${\mathbb{R}}^n$, inscribed polygonal sums, arc length as their supremum, and rectifiability. Its reversal is ${\gamma }^{-}(t):=\gamma (a+b-t)$. It is closed when $\gamma (a)=\gamma (b)$.

For paths $\alpha ,\beta :[0,1]\to {\mathbb{R}}^n$ with $\alpha (1)=\beta (0)$, their concatenation is

$(\alpha \ast \beta )(t):=\{\begin{array}{ll}\alpha (2t),& 0\le t\le \frac{1}{2},\\ \beta (2t-1),& \frac{1}{2}\le t\le 1.\end{array}$

When $a<b$ and $c<d$, if $h:[c,d]\to [a,b]$ is a continuous piecewise-$C^1$ bijection whose derivative has a fixed nonzero sign on every smooth piece, then $\gamma \circ h$ is an oriented piecewise-$C^1$ reparametrization. It is orientation-preserving when $h^{\mathrm{\prime }}>0$ and orientation-reversing when $h^{\mathrm{\prime }}<0$. Bijectivity excludes multiple coverings. Constant paths are allowed, although they are not regular; their speed-integral length is zero by A continuous piecewise-$C^1$ path is rectifiable and its length is the sum of the speed integrals over its pieces.