Where should I put my mirror?
Pyth - 19 17 15 13 bytes
Thanks to @FryAmTheEggman for saving me two bytes.
ARRGH the special case for no answer. Solved that!
e_I#jL\|cL2.:

Test Suite.
e                Last element in list, this works out to longest one
  _I             Invariance under reverse, this detect palindrome
   #             Filter
   jL            Map join
    \|           By &quot;|&quot;
    cL2          Map chop in two pieces
     .:Q)        Substrings. Implicit Q). ) makes it do all substrings.