I believe I have the answer:

>! $\begin{bmatrix} \begin{bmatrix} 3 \ 5 \ 8 \\ 7 \ 4 \ 6 \\ 1 \ 2 \ 9 \end{bmatrix} \begin{bmatrix} 2 \ 4 \ 6 \\ 9 \ 1 \ 3 \\ 7 \ 8 \ 5 \end{bmatrix} \begin{bmatrix} 9 \ 7 \ 1 \\ 5 \ 2 \ 8 \\ 3 \ 4 \ 6 \end{bmatrix} \\ \begin{bmatrix} 6 \ 7 \ 5 \\ 4 \ 9 \ 3 \\ 8 \ 1 \ 2 \end{bmatrix} \begin{bmatrix} 1 \ 9 \ 4 \\ 8 \ 5 \ 2 \\ 6 \ 3 \ 7 \end{bmatrix} \begin{bmatrix} 2 \ 8 \ 3 \\ 6 \ 1 \ 7 \\ 4 \ 5 \ 9 \end{bmatrix} \\ \begin{bmatrix} 2 \ 3 \ 7 \\ 5 \ 8 \ 1 \\ 9 \ 6 \ 4 \end{bmatrix} \begin{bmatrix} 4 \ 6 \ 1 \\ 3 \ 2 \ 9 \\ 5 \ 7 \ 8 \end{bmatrix} \begin{bmatrix} 8 \ 9 \ 5 \\ 7 \ 6 \ 4 \\ 1 \ 3 \ 2 \end{bmatrix} \end{bmatrix}$

The diagonals are like the rows and columns.

The anti-diagonal is in best order.  

>! These two clues are pretty straight forward, the diagonals can only have one of each number [1-9] in them and the anti-diagonal has them in the $1,2,3,...,9$ or $9,8,7,...,1$ order.

About the subsquares, you should know the following:

Upper left: Diagonal hints are squared unless left. There are odd hints on the left, and even a middle row. The perfect hint is right in the middle. There is an odd number of corner hints; that number is one of them.

>! The diagonal numbers are squares except for the left one so the middle and bottom right are 4 or 9.
>! The left column has only odd numbers.
>! 6 (the lowest perfect number) is in the middle right.
>! there are 3 odd numbers in the corners and one of them is the 3.

Upper middle: The first is even prime, the rest is unknown.

>! The top left is a 2.

Upper right: Five hints, like dots on dice. There is even a diagonal. It starts squared, with a perfect end.

There are 5 hints (3 of them mentioned in the anti-diagonal) So the top left is either 4 or 9 and the bottom right is 6.

Middle left: In that position is the smallest composite. The rest is silence.

>! That position is the middle left. the smallest composite number is 4.

Center: A wizard did it.

>! It's a magic square. (@Maria Deleva)

Middle right: The one in the middle right you have to figure out yourself. Or rather, you have to figure out everything else.

>! anything goes, again.

Lower left: Everything has already been said.

>! Not really a hint.

Lower middle: The square is even in the center. That's all.

>! The center square is a 2,4,6 or 8.

Lower right: The diagonal is going down. The numbers are multiple of their number.

>! Not entirely sure about this one.

Once I figured out what went where (with a little trial and error) I tossed it into a sudoku solver, as it involved using solving techniques I'm not super familiar with, and got my above answer.