= Solution
For a unit-speed curve with nonzero <curvature of a space curve> $\kappa$, define the <Frenet frame>
$$
T=\gamma',
\qquad N=\frac{T'}{\kappa},
\qquad B=T\times N.
$$
The <torsion of a space curve> is
$$
\boxed{\tau=-B'\mathbin\cdot N}
$$
with this choice of convention.
For the requested example, take the unit-speed circle
$$
\gamma_1(s)=(\cos s,\sin s,0)
$$
and the unit-speed <circular helix>
$$
\gamma_2(s)=
\left(\frac12\cos(\sqrt2s),
\frac12\sin(\sqrt2s),
\frac{s}{\sqrt2}\right).
$$
Both have the constant nonzero curvature $\kappa=1$, but their torsions are respectively $0$ and $1$. A <proper Euclidean motion of Euclidean three-space> preserves torsion, so no such rigid motion relates the two curves.
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