I know what $M^{-1}TM$ intuitively means. Suppose $T$ is the transformation matrix related with $\{\beta_i\}$ basis and $M$ is the transition matrix from $\{\alpha_i\}$ to $\{\beta_i\}$ then $$\underbrace{\left[M^{-1}\right]_\beta^\alpha\quad T\quad M_\alpha^\beta}_{\text{same transfromation matrix to $\{\alpha_i\}$ basis}}$$
So When even I saw any decomposition like that $M^{-1}TM$ then it give me the same intuition. But I was wondering is there also an intuitively meaning of
$$\underbrace{M^{T}\quad T\quad M}_?$$
If any other meaning do exist include them. I want to see them also.