= Solution
Under the specified passive change of <orthonormal basis>, each Cartesian <tensor> index acquires one copy of $L$. Thus a <tensor> of order $n$ transforms as
$$
\boxed{S'_{a_1\cdots a_n}=L_{a_1b_1}\cdots L_{a_nb_n}S_{b_1\cdots b_n}.}
$$
Repeated old-basis indices are summed from one to three. <Orthogonality> of the new basis gives $LL^T=I$. Taking <determinants> yields $(\det L)^2=1$, hence $\boxed{\det L=+1\text{ or }-1}$. The positive case preserves orientation and the negative case reverses it.
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