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Let $b=f(0)$ and $g(x)=f(x)-b$. Distance preservation and polarization give $⟨g(x),g(y)⟩=⟨x,y⟩$. The vectors $g(e_1),g(e_2),g(e_3)$ form an orthonormal basis, and $⟨g(x),g(e_i)⟩=x_i$. Hence $g(x)={\sum }_i{x}_{i}g(e_i)=Qx$ for an orthogonal matrix $Q$. Every Euclidean isometry is therefore $f(x)=Qx+b$.

To decompose $Q$, if $Q{e}_1\ne e_1$ reflect in the origin plane normal to $Q{e}_1-e_1$. This reflection sends $Q{e}_1$ to $e_1$. The remaining orthogonal transformation fixes $e_1$ and acts on its two-dimensional orthogonal complement; repeat with $e_2$, and then on the last one-dimensional complement. At most three plane reflections are needed. A translation $x\mapsto x+b$ is itself a product of two reflections in parallel planes: for $\mathbf{n}=b/|b|$, reflecting first in $\mathbf{n}\cdot x=0$ and then in $\mathbf{n}\cdot x=|b|/2$ gives that translation. Thus every isometry is a finite composition of plane reflections, the finite reflection decomposition of a Euclidean isometry.

For an isometry fixing zero, the preceding construction gives $N\le 3$. A plane reflection's linear part has $\mathrm{rank}(I-R)=1$. The identity $I-AB=(I-A)+A(I-B)$ implies that a product of $m$ such linear parts has rank of $I-Q$ at most $m$, even if the reflecting planes are affine. For central inversion $Q=-I$, this rank is three. The reflection length of a Euclidean orthogonal map therefore gives

For the regular tetrahedron, center it at zero. Every permutation of its four vertices determines an isometry, and the orientation-reversing ones correspond to odd permutations in $S_4$: six transpositions and six four-cycles. Each transposition is a plane reflection fixing the two unswapped vertices; its plane contains their edge and the midpoint of the opposite edge. These give six distinct mirror planes.

The other six are rotoreflections: rotate by $+\pi /2$ or $-\pi /2$ about an axis through the midpoints of a pair of opposite edges, and then reflect in the plane through the center perpendicular to that axis. There are three axes and two senses. For vertices $(1,1,1),(1,-1,-1),(-1,1,-1),(-1,-1,1)$, the map $(x,y,z)\mapsto (-y,x,-z)$ is one such four-cycle; the other axes and inverse rotations give all six. Hence the complete orientation-reversing tetrahedral symmetry list is six mirror reflections and six quarter-turn rotoreflections.