= Solution
At a fixed initial parameter choose an <orthogonal> map $Q$ sending $\widetilde T$ to $T$, $\widetilde N$ to $N$ and $\widetilde B$ to $-B$. Its determinant is $-1$. The transformed curve $Q\widetilde\alpha$ has <Frenet frame> $(Q\widetilde T,Q\widetilde N,-Q\widetilde B)$, so its <curvature> is $k$ and its torsion is $-\widetilde\tau=\tau$.
Both its frame and the frame of $\alpha$ solve the same linear Frenet system with the same initial data. The uniqueness theorem for a linear ordinary differential system with continuous coefficients makes the frames identical on the whole interval. In particular $Q\widetilde\alpha'=\alpha'$, so their difference is a constant vector $v$. Thus
$$
\boxed{\alpha=Q\widetilde\alpha+v,\qquad Q\in O(3),\quad\det Q=-1.}
$$
The orientation-reversing <rigid motion> accounts for the reversal of torsion.
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