As explained in my article Nine-Square with Palindromic Diagonal (Word Ways May 2002), the SW-NE diagonal must be palindromic. Such squares are rare (say, as common as ten-squares). Distinctly rarer is a nine-square such as follows, which has a frequency I estimate to be around 1 in 300,000 nine-squares.
"A Nine-Square With A Non-Palindrome Diagonal,"
Word Ways: Vol. 35
, Article 19.
Available at: https://digitalcommons.butler.edu/wordways/vol35/iss4/19