Euclidean isometries of $\mathbb{R}^3$ vs isometries of surfaces

Euclidean isometries in $\mathbb{R}^3$ are compositions of a translation and an orthogonal transformation. Each Euclidean isometry is a surface isometry that preserves length of rectifiable curves, but the converse is not true. A length-preserving isometry between $\mathbb{R}^3$ surfaces in $S_1$ and $S_2$ is a bijection that preserves dot products of tangent vectors.

For local isometries there is plenty of example, but I need an example of surface isometry that is not an Euclidean isometry.

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Any example that you can imagine making of paper will do. Paper doesn't stretch or shrink, it only bends, so if you can turn some paper surface into another paper surface, they're related by a surface isometry. For instance, if you have an infinite cylinder, you can squash it so as to make the cross section elliptical, which obviously changes the distances of the points in $\mathbb R^3$, but doesn't change their distances on the surface.