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Palindromic Sentences on left flexagon: rats live on no evil star; poor Dan is in a droop; and sit on a potato pan Otis.
Palindromic Sentences on right flexagon: A Santa stops pots at NASA; Draw no sore zeros on Ward; Draw no party booby trap onward.
Webster defines a palindrome as: "a word, line, verse, number, sentence, etc., reading the same backward as forward."

The above shows one face of a palindromic flexagon that is using six words that can be used to say the same sentence both backwards and forwards. I suspect that this could be easily expanded to other variations involving palindromes. A nice challenge would be to find a combination of words that would make reasonable sentences on each face of a flexagon. For now, to help us create more palindrome flexagons like the above left palindrome flexagon, we can select from the huge list of palindrome sentences at Mockok.com Palindrome Collection (more than 5000).