A [i]palindromic table[/i] is a $3 \times 3$ array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below. 
\[ \begin{array}[h]{ccc}
  O & M & O \\
  N & M & N \\
  O & M & O
\end{array} \]
How many palindromic tables are there that use only the letters $O$ and $M$?  (The table may contain only a single letter.)

[i]Proposed by Evan Chen[/i]
To determine the number of palindromic tables that use only the letters \( O \) and \( M \), we need to analyze the constraints imposed by the palindromic property on a \( 3 \times 3 \) array.

1. **Identify the positions in the table:**
   \[
   \begin{array}{ccc}
   a & b & c \\
   d & e & f \\
   g & h & i \\
   \end{array}
   \]

2. **Apply the palindromic constraints:**
   - Each row must read the same forwards and backwards.
   - Each column must read the same forwards and backwards.

   This gives us the following constraints:
   - First row: \( a = c \)
   - Second row: \( d = f \)
   - Third row: \( g = i \)
   - First column: \( a = g \)
   - Second column: \( b = h \)
   - Third column: \( c = i \)

3. **Simplify the table based on these constraints:**
   \[
   \begin{array}{ccc}
   a & b & a \\
   d & e & d \\
   a & b & a \\
   \end{array}
   \]

4. **Determine the independent variables:**
   - \( a \) (corner element)
   - \( b \) (middle element of the first row and column)
   - \( d \) (middle element of the second row and third column)
   - \( e \) (center element)

   These four variables are independent and can each be either \( O \) or \( M \).

5. **Calculate the total number of possible tables:**
   Each of the four independent variables has 2 choices (either \( O \) or \( M \)). Therefore, the total number of palindromic tables is:
   \[
   2 \times 2 \times 2 \times 2 = 2^4 = 16
   \]

Conclusion:
\[
\boxed{16}
\]