Question regarding the similarity of an invertible matrix with its inverse . 
Find the set $S$ of all possible $n×n$ invertible matrices $M$ such that $M$ is similar to $M^{-1}$ .

My approach
Actually, I was thinking about this problem when I came across a theorem stating that $\mathbf {every}$ square matrix is $\mathbf {similar}$ to its $\mathbf {transpose} .$ 
Obviously, for the trivial cases if $M = M^{-1}$ or $M^{-1} = M^t$ , then $S$ consists of all possible orthogonal matrices and matrices of order $2$ over $\mathbb R$ or $\mathbb C$ as similarity over one holds $\mathbf {iff}$ over another . 
Also, I thought that if $M \in S$, then if all  possible eigenvalues of $M$ are $\{ \lambda_j : 1 \leq j \leq k\}$ where each $\lambda_j$ appears $n_j$ times , then all  possible eigenvalues of $M^{-1}$ are $\{ 1/\lambda_j : 1 \leq j \leq k\}$ . 
Also, if $TMT^{-1} = M^{-1}$, then $M$ and $T^2$ commutes .
 A: I answer the question for $\mathbf C$. Consider the Jordan canonical form (JCF) $\mbox{bl.diag.}(\lambda_jI_j+N_j,j=1,...r)$ of $M$, where bl.diag. means "block diagonal matrix of...", $I_j$ is the identity matrix of size $n_j$, $N_j$ the matrix of size $n_j$ consisting of zeros except on the first
superdiagonal and where $\lambda_j\neq0$ are the eigenvalues of $M$. They are not necessarily distinct in this way to write the JCF.
The JCF of $M^{-1}$ is then $\mbox{bl.diag.}(\lambda_j^{-1}I_j+N_j,j=1,...r)$.
$M$ and $M^{-1}$ are similar if and only if their JCFs agree except for a permutation of the diagonal blocks. 
Therefore the set of all invertible matrices similar to its inverse consists of all matrices having a JCF $\mbox{bl.diag.}(\lambda_jI_j+N_j,j=1,...r)$ such that $\mbox{bl.diag.}(\lambda_j^{-1}I_j+N_j,j=1,...r)$ is equal to $\mbox{bl.diag.}(\lambda_jI_j+N_j,j=1,...r)$ except for a permutation.
In particular, the set $E$ of its eigenvalues must be stable by the mapping $x\mapsto1/x$, but this is not enough.
The answer for $\mathbf R$ is the same. In this case only the JCF of $M$ has an additional property: $\mbox{bl.diag.}(\overline{\lambda_j}I_j+N_j,j=1,...r)$ is equal to $\mbox{bl.diag.}(\lambda_jI_j+N_j,j=1,...r)$ except for a permutation of the diagonal blocks, where $\overline a$ means the complex conjugate of $a$.
