What are the matrices that you can write in the form $X \odot X^{-T}$, for a complex square matrix $X$, where $X^{-T}$ is the inverse of the complex transpose (not conjugate) and $\odot$ is the Hadamard (component-by-component) product?

In the $2\times 2$ case, you get the group of matrices in the form \begin{bmatrix}a & b\\ b & a \end{bmatrix}, which are the hyperbolic rotations $\oplus$ a multiplicative factor. In larger dimension, one sees that all the obtained matrices have the vector of all ones as both a right and left eigenvector. Is this the only restriction? Do the matrices that we obtain form a group? Is this problem known and studied?

Origin: motivated from this MO question.