Eigenvectors for a sum of diagonal and anti-diagonal matrices

Your X-matrices are actually a pretty well known family in disguise: block diagonal matrices with blocks $2\times2$. Go read that paragraph in Wikipedia, it is pretty short, and whatever you need from it is shorter yet:

The eigenvalues and eigenvectors of ${A}$ are simply those of $A_{1}$ and $A_{2}$ and ... and $A_{n}$ (combined).

Indeed, in an X-matrix, $x_1$ interacts only with $x_n$, $x_2$ with $x_{n-1}$, and so on. Why wouldn't you reorder your basis vectors so as to put the interacting ones next to each other? With your $5\times5$ example, this implies reordering the basis as $(x_1,x_5,x_2,x_4,x_3)$. The matrix which does that is $$P=\begin{pmatrix}1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 1& 0& 0\end{pmatrix}$$ Now apply that to your matrix $X$ and get $$PXP^T=\begin{pmatrix}1& 6& 0& 0& 0\\ 9& 5& 0& 0& 0\\ 0& 0& 2& 7& 0\\ 0& 0& 8& 4& 0\\ 0& 0& 0& 0& 3 \end{pmatrix}$$ Q.e.d.