Gram mates, sign changes in singular values, and isomorphism

We study distinct $(0,1)$ matrices $A$ and $B$, called $\mathit{\text{Gram}}\mathit{\text{mates}}$, such that $A{A}^T=B{B}^T$ and $A^{T}A=B^{T}B$. We characterize Gram mates where one can be obtained from the other by changing signs of some positive singular values. We classify Gram mates such that the rank of their difference is at most $2$. Among such Gram mates, we further produce equivalent conditions in order that one is obtained from the other by changing signs of at most $2$ positive singular values. Moreover, we provide some tools for constructing Gram mates where the rank of their difference is more than $2$. Finally, we characterize non-isomorphic Gram mates whose difference is of rank $1$ with some extra conditions.