Of course, after Gov. Sarah Palin, we should have known that "palin" stood for "backward."
Then there are palindromic pairs, words which are mirror images of each other. Put together, the two words form a palindrome. Or, reversed in order, the words form a palindrome. For example, a palindrome can be formed from "bat tab" or "tab bat."
In constructing palindromic phrases, these can be useful. For example, if we refer to a palindromic pair as A(1) and A(-1), and any palindromic word as Z(1), a palindromic phrase can be constructed out of A(1)+A(-1) or A(1)+Z(1)+A(-1). For example, we could make a palindromic phrase out of "rail liar" or "rail denned liar". Of course, there are limitless combinations of palindromic words and palindromic pairs which can form palindromic phrases. The challenge, which was not met in this paragraph's examples, is to form phrases which make some kind of sense.
Of course, please note that most palindromic phrases are not composed of phrases with one word in the middle between palindromic pairs. For example, "O had I nine more hero-men in Idaho!" shows no such symmetry.
The source for much of this a London Times article. Of course, we welcome other sources, such as you. Contact us with your palindromes.