Show that the matrix $B = \left(\begin{array}{cccc} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & \frac{1}{2} & 1 \\ 0 & 0 & 0 & \frac{1}{2} \end{array}\right)$ is not similar to its inverse (although its characteristic polynomial $(X-2)^2\left(X-\frac{1}{2}\right)^2$ is reciprocal).
One may determine the eigenspaces of $B$ and $B^{-1}$ for the eigenvalue 2.