 Solution

A column of coordinates transforms as ${\mathbf{x}}^{\prime }=U^{†}\mathbf{x}$, because the columns of $U$ are the new orthonormal basis vectors expressed in the old coordinates. A symmetric unitary choice isIts first and third columns are orthogonal eigenvectors of the $xz$ block, with eigenvalues $-1$ and $+1$ respectively; its second column is the $y$ axis with eigenvalue $3$. Direct multiplication checks $U^{†}U=I$, while the displayed entries give $U^T=U$. The symmetry here means transpose symmetry, not Hermitian symmetry.