= Solution
Use the given <torus> of independent rotations in $n$ mutually <orthogonal> two-planes and a fixed final axis. A <normalizer> element permutes the two-plane rotation characters, and on each plane either preserves or reverses its <orientation>. Its induced action therefore permutes the $n$ angle coordinates and changes their signs independently.
Every such signed permutation is realized. Permutations of two-dimensional blocks have <determinant> one. A reflection in any chosen plane reverses that angle; reflecting the remaining one-dimensional axis as necessary restores total <determinant> one. The kernel consists precisely of independent plane rotations: commuting with a generic <torus> element forces preservation of its distinct rotation planes, and the <determinant> condition then fixes the final axis. Thus the kernel is $T$ and
$$
\boxed{W\cong(C_2)^n\rtimes S_n,\qquad |W|=2^nn!}.
$$
This is the signed permutation <group>, the <Weyl group> of type $B_n$.
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