Karin Anderson suggests "if you wish to write a Palindrome yourself look at the example on Shadow Poetry , decide on a  theme, and begin with a phrase or small group of words that read both ways and build on them gradually. It will take time, but it is a challenge and is well worth it when the final result is produced and others say 'that is so clever'".

**A palindrome is a word, phrase, number or other sequence of units that can be read the same way in either direction (the adjustment of punctuation and spaces between words is generally permitted). Composing literature in palindromes is an example of constrained writing. The word "palindrome" was coined from Greek roots pálin (πάλιν; "again") and drómos (δρóμος; "way, direction") by English writer Ben Jonson in the 1600s. The actual Greek phrase to describe the phenomenon is karkinikê epigrafê (καρκινική επιγραφή; crab inscription), or simply karkiniêoi (καρκινιήοι; crabs), alluding to the backward movement of crabs, like an inscription which can be read backwards.

Palindromes date back at least to 79 AD; they existed in Hebrew,  ancient Sanskrit, Latin and many other languages (Preminger***).

Wikipedia cites various types of Palindromes including Characters, Phrases, Famous Quotations, Names, Words and lines.

Characters:
The most familiar palindromes, in English at least, are character-by-character: the written characters read the same backwards as forwards. Some examples of palindromic words: civic, radar, level, rotator, rotor, kayak, reviver, racecar, and redder.

Phrases:
Palindromes often consist of a phrase or sentence ("Go hang a salami I'm a lasagna hog.", "Was it a rat I saw?", "Step on no pets", "Sit on a potato pan, Otis", "Lisa Bonet ate no basil", "Satan, oscillate my metallic sonatas", "I roamed under it as a tired nude Maori", "Yo banana boy", "Rise to vote sir", or the exclamation "Dammit, I'm mad!"). Punctuation, capitalization, and spacing are usually ignored, although some (such as "Rats live on no evil star") include the spacing.[2]

Famous quotations:
Three famous English palindromes are "Able was I ere I saw Elba"[3] (which is also palindromic with respect to spacing), "A man, a plan, a canal, Panama!",[4] and "Madam, I'm Adam".

Names:
Some people have names that are palindromes. Lon Nol (1913–1985) was Prime Minister of Cambodia. Nisio Isin is a Japanese novelist and manga writer, whose real name (西尾 維新, Nishio Ishin) is a palindrome when romanized using Kunrei-shiki or Nihon-shiki (it is often written as NisiOisiN to emphasize this). Some changed their name in order to be a palindrome (one example is actor Robert Trebor), while others were given a palindromic name at birth (such as philologist Revilo P. Oliver and Korean-American Mike Kim).[5] Palindromic names are very common in Finland. Examples include Emma Lamme, Sanna Rannas, Anni Linna and Asko Oksa. "Stanley Yelnats" is the name of a character in "Holes", a 1998 novel and 2006 film.

Words:
Some palindromes use words as units rather than letters. Examples are "Fall leaves after leaves fall", "You can cage a swallow, can't you, but you can't swallow a cage, can you?", "First Ladies rule the State and state the rule: ladies first" and "Girl, bathing on Bikini, eyeing boy, sees boy eyeing bikini on bathing girl". The command "Level, madam, level!", composed only of words that are themselves palindromes, is both a character-by-character and a word-by-word palindrome.

Lines:
Still other palindromes take the line as the unit. The poem Doppelgänger, composed by James A. Lindon, is an example.

The dialogue "Crab Canon" in Douglas Hofstadter's Gödel, Escher, Bach is nearly a line-by-line palindrome. The second half of the dialog consists, with some very minor changes, of the same lines as the first half, but in reverse order and spoken by the opposite characters (i.e., lines spoken by Achilles in the first half are spoken by the Tortoise in the second, and vice versa). In the middle is a non-symmetrical line spoken by the Crab, who enters and spouts some nonsense, apparently triggering the reversal. The structure is modeled after the musical form known as crab canon, in particular the canon a 2 cancrizans of Johann Sebastian Bach's The Musical Offering.

The Palindrome has also been used in Classical and contemporary music.

Classical music:
Joseph Haydn's Symphony No. 47 in G is nicknamed "the Palindrome". The third movement, minuet and trio is a musical palindrome. This clever piece goes forward twice and backwards twice and arrives back at the same place.

W.A. Mozart's Scherzo-Duetto di Mozart is played by one violinist as written and the second with the same music inverted.

The interlude from Alban Berg's opera Lulu is a palindrome, as are sections and pieces, in arch form, by many other composers, including James Tenney, and most famously Béla Bartók. George Crumb also used musical palindrome to text paint the Federico Garcia Lorca poem "¿Porque nací?", the first movement of three in his fourth book of Madrigals. Igor Stravinsky's final composition, The Owl and the Pussy Cat, is a palindrome.

Contemporary music:
In 1975, the Swedish pop group ABBA had a hit single titled "SOS", a unique occasion in which a song's title and the name of its recording artist are both palindromes.

The title track of the 1992 album entitled UFO Tofu by Béla Fleck and the Flecktones is said by its composer to be a musical palindrome.

Shadow Poetry cites contemporary poetry examples.

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* "palindrome." Collins English Dictionary - Complete & Unabridged 10th Edition. HarperCollins Publishers. 01 Aug. 2010. <Dictionary.com http://dictionary.reference.com/browse/palindrome>.

**"Palindrome." Wikipedia: The Free Encyclopedia. 01 Aug. 2010. <en.wikipedia.org http://en.wikipedia.org/wiki/Palindrome>.

***"PALINDROME".  The New Princeton Encyclopedia of Poetry and Poetics. Alex Preminger and T.V.F. Brogan, editors.  New York: MJF Books, Princeton University Press, 1993, p. 874.