6. In a regular 2008-gon, the number of diagonals that are not parallel to any side is ( ).
(A) 2008
(B) $1004^{2}$
(C) $1004^{2}-1004$
(D) $1004^{2}-1003$
6. C.

For a regular $2n$-sided polygon $A_{1} A_{2} \cdots A_{2 n}$, the number of diagonals is
$$
\frac{1}{2} \times 2 n(2 n-3)=n(2 n-3) \text { (diagonals). }
$$

To calculate the number of diagonals parallel to a side $A_{1} A_{2}$:
Since $A_{1} A_{2} \parallel A_{n+1} A_{n+2}$, the endpoints of the diagonals parallel to $A_{1} A_{2}$ can only be chosen from $2 n-4$ points, resulting in $n-2$ parallel lines. Therefore, the number of diagonals parallel to any side is $n(n-2)$. Consequently, the number of diagonals that are not parallel to any side is
$$
n(2 n-3)-n(n-2)=n(n-1) \text { (diagonals). }
$$