A herder has forgotten the number of cows she has, and does not want to count them all of them. She remembers these four facts about the number of cows: 
[list]
[*]It has $3$ digits.
[*]It is a palindrome.
[*]The middle digit is a multiple of $4$.
[*]It is divisible by $11$.
[/list]
What is the sum of all possible numbers of cows that the herder has?

$\textbf{(A) }343\qquad\textbf{(B) }494\qquad\textbf{(C) }615\qquad\textbf{(D) }635\qquad\textbf{(E) }726$
1. The number of cows is a three-digit number, so it can be represented as $\underline{a} \ \underline{b} \ \underline{a}$, where $a$ and $b$ are digits.
2. The number is a palindrome, which means it reads the same forwards and backwards. Therefore, the number can be written as $aba$.
3. The middle digit $b$ is a multiple of $4$. The possible values for $b$ are $0, 4,$ and $8$.
4. The number is divisible by $11$. For a number to be divisible by $11$, the alternating sum of its digits must be divisible by $11$. For the number $aba$, this alternating sum is $a - b + a = 2a - b$.

We need to find the values of $a$ and $b$ such that $2a - b$ is divisible by $11$ and $b$ is a multiple of $4$.

Let's consider the possible values of $b$:
- If $b = 0$, then $2a - 0 = 2a$ must be divisible by $11$. Since $a$ is a digit (0-9), there is no value of $a$ that makes $2a$ divisible by $11$.
- If $b = 4$, then $2a - 4$ must be divisible by $11$. We solve for $a$:
  \[
  2a - 4 = 11k \quad \text{for some integer } k
  \]
  \[
  2a = 11k + 4
  \]
  \[
  a = \frac{11k + 4}{2}
  \]
  Since $a$ must be a digit, we check for integer values of $k$:
  - For $k = 0$, $a = \frac{4}{2} = 2$
  - For $k = 1$, $a = \frac{15}{2} = 7.5$ (not an integer)
  - For $k = -1$, $a = \frac{-7}{2} = -3.5$ (not an integer)
  Therefore, $a = 2$ is the only valid solution for $b = 4$. The number is $242$.
- If $b = 8$, then $2a - 8$ must be divisible by $11$. We solve for $a$:
  \[
  2a - 8 = 11k \quad \text{for some integer } k
  \]
  \[
  2a = 11k + 8
  \]
  \[
  a = \frac{11k + 8}{2}
  \]
  Since $a$ must be a digit, we check for integer values of $k$:
  - For $k = 0$, $a = \frac{8}{2} = 4$
  - For $k = 1$, $a = \frac{19}{2} = 9.5$ (not an integer)
  - For $k = -1$, $a = \frac{-3}{2} = -1.5$ (not an integer)
  Therefore, $a = 4$ is the only valid solution for $b = 8$. The number is $484$.

The possible numbers of cows are $242$ and $484$. The sum of these numbers is:
\[
242 + 484 = 726
\]

The final answer is $\boxed{726}$