Consider a self-adjoint matrix $M$ that has block form

$$M = \begin{pmatrix} M_{11} & M_{12} \\ M_{12}^* & M_{11} \end{pmatrix}.$$

I am wondering if there exists any criterion to decide if this matrix can be transformed by some invertible matrix $T$

such that $$TMT^{-1} = \begin{pmatrix}0 & C \\ C^* & 0 \end{pmatrix}$$ for some suitable matrix $C?$

Notice that one restriction that $\begin{pmatrix}0 & C \\ C^* & 0 \end{pmatrix}$ already puts is that the spectrum of $M$ has to be symmetric with respect to zero as conjugation by $$\begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}$$ shows.

As a first step, one might ask when we can achieve a form

$$TMT^{-1} = \begin{pmatrix}0 & C \\ D & 0 \end{pmatrix}$$

where $C$ and $D$ are arbitrary matrices?