This is a socalled chiral symmetry. The restriction on the symmetry of the spectrum is the only restriction you need, you can then bring it to the desired off-diagonal form by a unitary transformation:
$$M=U\begin{pmatrix}\Lambda&0\\ 0&-\Lambda\end{pmatrix}U^\ast$$
$$\Rightarrow \Omega^\ast U^\ast MU\Omega =\begin{pmatrix}0&\Lambda\\  \Lambda&0\end{pmatrix},$$
for $\Omega=2^{-1/2}\begin{pmatrix}1&1\\ -1 &1\end{pmatrix}$.