Consider $N \times N$ matrices

$$A = \begin{bmatrix} 0 & 0 & \cdots & 0 & 1 \\ 1 & 0 & 0 & & 0 \\ \vdots & 1 & 0 & \ddots & \vdots \\ 0 & & \ddots & \ddots & 0 \\ 0 & 0 & \cdots &1 & 0 \\ \end{bmatrix}$$ and

$$B=\operatorname{diag}( \cos(2\pi\cdot 0/N),...,\cos(2\pi\cdot (N-1)/N)).$$

Does anybody know why the eigenvalues of $i(A+A^T)+B$ are invariant under 90° rotations?- Numerics seem to imply this. What I mean by this is that if $\lambda$ is an eigenvalue, then also $e^{i \frac{\pi}{2}} \lambda.$