This may be a stupid question, but it looks to me like the reduced residue systems modulo N are symmetrical about N/2; that is to say, that the there is the same number of integers not divisible by a factor of N that are smaller than N/2 as there are that are bigger. Is this a case, and is there literature I would need to reference in a paper that uses this fact?
Just to be clear, in this case, I use "Reduced Residue System" to refer to the system of numbers counted by the totient of N, i.e. those numbers smaller than N that are relatively prime with N.
This is certainly true. You would not need to cite any literature to use such a thing in a paper, as the proof follows immediately from the definition of divisibility (I wrote two below). If you really wanted to cite something, cite any introductory number theory book that explains divisibility and modular arithmetic.
For completeness, a pair of proofs.
Claim: If $m < n$ has $\gcd(m,n) = 1$, then $\gcd(n-m,n)=1$.
proof 1: $d|(n-m)$ and $d|n \iff d|m$ and $d|n$ by divisibility definitions. $\spadesuit$
proof 2: Let's use Bezout, as an exercise. Then $\gcd(n,m)=1 \implies$ there are integers $a,b$ s.t. $an + bm = 1$. But then $-b(n-m) + (a+b)n = 1$, so that $\gcd(n-m,n) = 1$ too. $\clubsuit$