How many pairs of letters are in the word ‘ OVERWHELMING’, each of which have as many letters between them (both forward and backward direction) as they have between them according to English alphabetical order?

To find the number of letter pairs in the word OVERWHELMING that have the same number of letters between them in the word as they do in the English alphabetical order (in both forward and backward directions), let us examine the position of each letter in the word.

Let us write down the word along with the numerical positions of each letter in the standard English alphabet:
O (15), V (22), E (5), R (18), W (23), H (8), E (5), L (12), M (13), I (9), N (14), G (7)

Now, we look for pairs of letters $L_1$ and $L_2$ at indices $i$ and $j$ (where $i<j$) such that the absolute difference in their alphabetical positions is equal to the distance between them in the word, which is $j-i$.

Let us test the pairs:

1. O (15) and R (18):
The letters in the word are: O V E R.
Distance in the word = 3.
Difference in alphabetical position = 18 - 15 = 3.
This is the first pair: (O, R).

2. E (5) and H (8):
The letters in the word are: E R W H.
Distance in the word = 3.
Difference in alphabetical position = 8 - 5 = 3.
This is the second pair: (E, H) (using the first 'E').

3. L (12) and M (13):
The letters in the word are next to each other: L M.
Distance in the word = 1.
Difference in alphabetical position = 13 - 12 = 1.
This is the third pair: (L, M).

4. L (12) and I (9):
The letters in the word are: L M I.
Distance in the word = 2 (in backward direction: I, J, K, L).
Difference in alphabetical position = 12 - 9 = 3 (Wait, let's verify: alphabetical order is I, J, K, L, which has 2 letters between them, so the difference is 3, but the distance in the word is 2. Thus, (L, I) is not a pair).
Let us check: E (5, seventh letter) and G (7, twelfth letter): distance is 5, alphabet difference is 2. No.
Let us check: I (9, tenth letter) and G (7, twelfth letter):
The letters in the word are: I N G.
Distance in the word = 2.
Difference in alphabetical position = 9 - 7 = 2.
This is the fourth pair: (I, G) (or backward direction: G, H, I).

5. N (14) and G (7):
Let us check: H (8, sixth letter) and L (12, eighth letter): distance is 2, alphabet difference is 4. No.
Let us check: E (5, third letter) and I (9, tenth letter): distance is 7, difference is 4. No.
Let us check: M (13) and I (9): distance is 1, difference is 4. No.
Let us check: V (22) and W (23):
The letters in the word are: V E R W.
Distance in the word = 3, difference is 1. No.
Let us check: E (5, third letter) and E (5, seventh letter): distance is 4, difference is 0. No.
Let us check: O (15, first letter) and W (23, fifth letter): distance is 4, difference is 8. No.
Let us check: M (13, ninth letter) and G (7, twelfth letter):
The letters in the word are: M I N G.
Distance in the word = 3 (backward direction: G, H, I, J, K, L, M, which has 5 letters between them, so difference is 6). No.
Let us check: E (5, third letter) and L (12, eighth letter):
The letters in the word are: E R W H E L.
Distance in the word = 5.
Difference in alphabetical position = 12 - 5 = 7. No.
Let us check: E (5, seventh letter) and I (9, tenth letter):
The letters in the word are: E L M I.
Distance in the word = 3.
Difference in alphabetical position = 9 - 5 = 4. No.
Let us check: E (5, seventh letter) and N (14, eleventh letter):
The letters in the word are: E L M I N.
Distance in the word = 4.
Difference in alphabetical position = 14 - 5 = 9. No.
Let us check: E (5, seventh letter) and G (7, twelfth letter):
The letters in the word are: E L M I N G.
Distance in the word = 5.
Difference in alphabetical position = 7 - 5 = 2. No.
Let us check: O (15) and N (14):
The letters in the word are: O V E R W H E L M I N.
Distance in the word = 10, difference is 1. No.
Let us check: N (14, eleventh letter) and G (7, twelfth letter): distance is 1, difference is 7. No.
Let us check: R (18, fourth letter) and N (14, eleventh letter): distance is 7, difference is 4. No.
Let us check: H (8, sixth letter) and M (13, ninth letter):
The letters in the word are: H E L M.
Distance in the word = 3.
Difference in alphabetical position = 13 - 8 = 5. No.
Let us check: H (8, sixth letter) and I (9, tenth letter):
The letters in the word are: H E L M I.
Distance in the word = 4.
Difference in alphabetical position = 9 - 8 = 1. No.
Let us check: H (8, sixth letter) and N (14, eleventh letter):
The letters in the word are: H E L M I N.
Distance in the word = 5.
Difference in alphabetical position = 14 - 8 = 6. No.
Let us check: H (8, sixth letter) and G (7, twelfth letter):
The letters in the word are: H E L M I N G.
Distance in the word = 6.
Difference in alphabetical position = 8 - 7 = 1. No.
Let us check: E (5, third letter) and M (13, ninth letter):
The letters in the word are: E R W H E L M.
Distance in the word = 6.
Difference in alphabetical position = 13 - 5 = 8. No.
Let us check: E (5, third letter) and N (14, eleventh letter):
The letters in the word are: E R W H E L M I N.
Distance in the word = 8.
Difference in alphabetical position = 14 - 5 = 9. No.
Let us check: E (5, third letter) and G (7, twelfth letter):
The letters in the word are: E R W H E L M I N G.
Distance in the word = 9.
Difference in alphabetical position = 7 - 5 = 2. No.
Let us check: O (15, first letter) and V (22, second letter): distance is 1, difference is 7. No.
Let us check: O (15, first letter) and I (9, tenth letter):
The letters in the word are: O V E R W H E L M I.
Distance in the word = 9.
Difference in alphabetical position = 15 - 9 = 6. No.
Let us check: V (22, second letter) and R (18, fourth letter): distance is 2, difference is 4. No.
Let us check: W (23, fifth letter) and R (18, fourth letter): distance is 1, difference is 5. No.
Let us check: W (23, fifth letter) and M (13, ninth letter): distance is 4, difference is 10. No.
Let us check: R (18, fourth letter) and L (12, eighth letter):
The letters in the word are: R W H E L.
Distance in the word = 4.
Difference in alphabetical position = 18 - 12 = 6. No.
Let us check: R (18, fourth letter) and M (13, ninth letter):
The letters in the word are: R W H E L M.
Distance in the word = 5.
Difference in alphabetical position = 18 - 13 = 5.
This is the fifth pair: (R, M) (or backward direction: M, N, O, P, Q, R).

Thus, the 5 pairs of letters satisfying the given condition are:
1. (O, R)
2. (E, H)
3. (L, M)
4. (I, G)
5. (R, M)

Therefore, the correct count is Five.