Congruence criteria for normal and conjugate-normal matrices

Complex $n\times n$ matrices $A$ and $B$ are said to be $T$-congruent if $B = S^T AS$ and $*$-congruent if $B = S^* AS$, where $S$ is an arbitrary nonsingular matrix. For several facts related to normal matrices and $*$-congruences, analogs in the theory of $T$-congruences, concerning conjugate-normal matrices, are found.