After considering this problem again, I believe that my answer is correct - 16.
Looking at Melody's answers, she states that AABB and BBAA are palindromes.
But after doing a little research, I discovered that this isn't quite correct. A palindromic number or numeral palindrome is a number that remains the same when its digits are reversed. See the description, here....
http://en.wikipedia.org/wiki/Palindromic_number
Notice, if we chose 5 and 2..... 5522 isn't the same as 2255. (They are different numbers)
The only two palindromes we can make here are 2552 and 5225.
To see this more clearly......note that we have 4 ways to choose the first number and 3 ways to choose the second number = 12. And the digits in the 3rd and 4th positions will just be the reverse of this order.
So....these 12 ways plus the 4 ways of choosing the same digit to occupy all four positions gives us only 16 possibilities.