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