permutation such that adding its location to each element creates a new permutation

Question

For even n, does there exist a permutation of the elements of Z_n such that adding the location in the permutation to each element of the permutation creates a new permutation?

For example if n is odd instead (n=5):
Consider the permutation 01234, if you add the location in the permutation to each element (the first element being at location 0) you get a new permutation 02413

I don't think this is possible for even n in general however I have only proven this true by exhaustion for a few even n. I would like to know a proof for all even n, or if no proof exists, I'd like a counter example.

Thanks

Here is another example:
<0,2,4,1,3> + <0,1,2,3,4> = <0,3,1,4,2>
notice the first last last vector all have different entries

Answer 1

If $n$ is even, then the sum of the elements of $\Bbb Z_n$ is $n/2$. So if you "add" two permutations of $\Bbb Z_n$, then you get a list of elements of $\Bbb Z_n$ whose sum is zero, and so this list cannot be a permutation.