1. Define the initial string and functions:
Let $T$ be the initial string of 250 letters, consisting of 125 letters $A$ and 125 letters $B$. Define $f_A(S)$ as the number of $A$'s in even positions in a string $S$, and $f_B(S)$ as the number of $B$'s in even positions.

2. Claim:
The value of $f_A(S)$ oscillates between $f_A(T)$ and $125 - f_B(T)$ under the defined move.

3. Proof of the claim:
- Consider a string $S$ and a resulting string $S_1$ after performing the operation on a certain substring of $S$.
- If a $B$ was in an odd position, reversing the substring sends this $B$ to an even position. Then, flipping the letters turns this $B$ into an $A$. Thus, every $B$ in an odd position is converted into an $A$ in an even position.
- The number of $B$'s in odd positions is $125 - f_B(S)$.
- Therefore, after the operation:
$$f_A(S_1) = 125 - f_B(S)$$
- Performing the operation again on $S_1$ results in:
$$f_A(S_2) = 125 - f_B(S_1) = 125 - (125 - f_A(S)) = f_A(S)$$
- This proves that $f_A(S)$ oscillates between $f_A(T)$ and $125 - f_B(T)$.

4. Reversing the string:
- When we reverse the string $T$, every $A$ in an even position goes to an odd position. Let $T'$ be the reversed string of $T$.
- Therefore:
$$f_A(T') = 125 - f_A(T)$$

5. Condition for the string to be the same as the initial string but reversed:
- We need $125 - f_A(T) = f_A(T)$, which implies:
$$2f_A(T) = 125$$
- This is impossible since 125 is odd, and $f_A(T)$ must be an integer.
- Alternatively, we need $125 - f_A(T) = 125 - f_B(T)$, which implies:
$$f_A(T) = f_B(T)$$
- Since there are 125 even positions in the 250-letter string, this means:
$$f_A(T) = 125 - f_A(T)$$
- This is also impossible for the same reason (125 is odd).

6. Conclusion:
- It is clear that one can never reverse the starting string for any ordering of the letters.

The final answer is False.