I am one of many siblings in a great family.
We are the most perfect of our race,
Because each of us has a clear worst enemy.
My foe - I know him well - is much like me,
But we continually fight each other;
We never hide our hatred for each other.
Wave at my eldest brother and he becomes me.
When my dear cousin finds the end of her life,
She too can turn into me, in her gentle way.
My younger brother thinks too highly of himself,
So smooth and perfect that he'll always outstrip me.
Often he even steals my words, confusing all concerned.
What am I?
1) As @KeithS worked out, the answer is something in pure maths, and more abstract than anything anyone has tried so far. You need to know a bit of undergraduate-level maths, at the level of basic topology and group theory.
2) Every line has a thought-out meaning, so you need to find a solution that matches everything from "we never hide our hatred for each other" to "he'll always outstrip me". I could give you a word for each of the "family", "siblings", "cousin", "eldest brother", and "younger brother". About the only red herring is "in her gentle way", which is just a nod to Thomas Hardy.
The answer is a mathematical object which someone has mentioned (not proposing it as an answer, but just mentioning it in passing) in one of the responses below. I won't say which one!
I am the only one who can go directly to Leipzig from a well-policed city.