= Solution
Use the <Cartesian second-rank tensor> transformation convention $T'=RTR^T$, where $R$ is the orthogonal matrix converting components to the new <orthonormal basis>. The coordinate half-turns are
$$
R_x=\operatorname{diag}(1,-1,-1),\quad
R_y=\operatorname{diag}(-1,1,-1),\quad
R_z=\operatorname{diag}(-1,-1,1).
$$
For a diagonal rotation $R=\operatorname{diag}(s_1,s_2,s_3)$, invariance gives $t_{ij}=s_is_jt_{ij}$, with no summation in this equation. For each $i\ne j$, one of the listed half-turns makes $s_is_j=-1$, so $t_{ij}=-t_{ij}=0$. Thus
$$
\boxed{T=\operatorname{diag}(t_{11},t_{22},t_{33}).}
$$
This is <coordinate half-turn invariance of a second-rank tensor>. It does not force equal diagonal entries; invariance under all rotations would be the stronger <isotropic second-rank tensor> condition.
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