Find the sum of all primes that can be written both as a sum of two primes and as a difference of two primes.

[i]Anonymous Proposal[/i]
1. **Identify the conditions for the primes:**
   - A prime \( p \) must be expressible as both a sum of two primes and a difference of two primes.
   - For \( p \) to be a sum of two primes, one of the primes must be 2 (since the sum of two odd primes is even and greater than 2, hence not prime).
   - For \( p \) to be a difference of two primes, one of the primes must be 2 (since the difference of two odd primes is even and greater than 2, hence not prime).

2. **Set up the equations:**
   - Let \( p = q_1 + 2 \) where \( q_1 \) is an odd prime.
   - Let \( p = q_2 - 2 \) where \( q_2 \) is an odd prime.

3. **Equate the two expressions for \( p \):**
   \[
   q_1 + 2 = q_2 - 2
   \]
   \[
   q_2 = q_1 + 4
   \]

4. **Analyze the possible values of \( p \):**
   - Since \( p \) must be a prime, we need to check the values of \( p \) that satisfy both conditions.
   - Suppose \( p > 5 \). Then \( q_1 \) and \( q_2 \) must be greater than 3 (since they are odd primes).

5. **Consider modulo 3 analysis:**
   - If \( p \equiv 1 \pmod{3} \), then \( q_2 \equiv 0 \pmod{3} \), which is a contradiction since \( q_2 \) is a prime greater than 3.
   - If \( p \equiv 2 \pmod{3} \), then \( q_1 \equiv 0 \pmod{3} \), which is a contradiction since \( q_1 \) is a prime greater than 3.

6. **Conclude that \( p \leq 5 \):**
   - The only primes less than or equal to 5 are 2, 3, and 5.
   - Check each case:
     - \( p = 2 \): Cannot be written as the sum of two primes.
     - \( p = 3 \): Cannot be written as the sum of two primes.
     - \( p = 5 \): Can be written as \( 5 = 3 + 2 \) and \( 5 = 7 - 2 \).

7. **Verify the solution:**
   - The only prime that satisfies both conditions is \( p = 5 \).

The final answer is \(\boxed{5}\).