10. C.

For different permutations of abcd, if a is in the first position, then the second position can only be c or d. Let's assume it is c. At this point, there is no letter to place in the third position. If we place b, then cb are adjacent; if we place d, then cd are adjacent, which is a contradiction. Therefore, a cannot be in the first position.
Similarly, d cannot be in the first position either.
If b is in the first position, then the second position can only be d, the third position a, and the last position c, i.e., bdac.

Similarly, cadb also fits. Therefore, there are two permutations that meet the conditions.
10. For the letter arrangement abcd, there are ( ) different rearrangements such that no two letters that were adjacent in the original arrangement are adjacent in the rearrangement (for example, ab and ba should not appear in the rearrangement).
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4