Congratulations to dmg for getting the intended answer!
Here are my explanations for the clues.
The first 4 lines are about the word "palindrome" itself.
Within me you will find One of like heart and mind.
A friend is a person of like heart and mind. Pal is a synonym for friend, and "pal" is a substring of palindrome. (I thought about using a clue for "palin" instead - as in Sarah Palin - but I thought that would be too U.S.-centric.)
As well as an entity With inverted geography.
Countries have cities within them, but there is one example of a city (Rome) having a country (Vatican City) within it. "Rome" is a substring of palindrome.
The last 4 lines are about the meaning of the word palindrome.
While I represent symmetry
A palindrome by definition has symmetry.
Mirrors can show another identity.
If you hold a word up to a mirror, it will not look the same, and may form a different word. For example, the palindrome "bib" turns into "did" in a mirror.
But there will always be twins
The first letter of a palindrome is twins with the last letter. The second letter is twins with the second-to-last letter, and so on.
Who generate many grins.
Many people think palindromes are fun!
Finally, the singular form of "boobs" is a palindrome.