The correct answer is 1,992 different ways. 
Every F is either a corner F or a side F—standing next to a corner in its own square of F's. 
Now, FIED may be read from a corner F in 16 ways; therefore DEIF may be read into a corner F also in 16 ways; hence DEIFIED may be read through a corner F in 16 × 16 = 256 ways. 
Consequently, the four corner F's give 4 × 256 = 1,024 ways. 
Then FIED may be read from a side F in 11 ways, and DEIFIED therefore in 121 ways. 
But there are eight side F's; consequently these give together 8 × 121 = 968 ways. 
Add 968 to 1,024 and we get the answer, 1,992.

In this form the solution will depend on whether the number of letters in the palindrome be odd or even. 
For example, if you apply the word NUN in precisely the same manner, you will get 64 different readings; but if you use the word NOON, you will only get 56, because you cannot use the same letter twice in immediate succession (since you must "always pass from one letter to another") or diagonal readings, and every reading must involve the use of the central N.

The reader may like to find for himself the general formula in this case, which is complex and difficult. I will merely add that for such a case as MADAM, dealt with in the same way as DEIFIED, the number of readings is 400.