Suppose $A\in K^{n\times n}$ is diagonalizable via an involutory (or self-inverse) matrix, i.e. there exist a diagonal $D\in K^{n\times n}$ and an involutory transformation $T\in K^{n\times n}$ (i.e. $T^{-1}=T$) such that $$A=TDT^{-1}(=TDT)$$
What would be a characterization of such matrices, in particular for $K=\mathbb{C}$ (and $K=\mathbb{R}$)?