Well, whether it's obvious or not, I think it's in the paper. Immediately under equation (1.1) of http://arxiv.org/abs/1603.03720 http://arxiv.org/abs/1603.0372

Well, whether it's obvious or not, I think it's in the paper. Immediately under equation (1.1) of http://arxiv.org/abs/1603.03720 http://arxiv.org/abs/1603.03720, the authors are discussing the second correction term to the distribution of primes mod q, and they state: "We can also show that c2(q; (a, b)) = c2(q; (−b,−a)) for any two reduced residue classes a and b (mod q)." I'm not 100% certain this is responsible for the phenomenon that we're seeing, but it seems exceedingly likely. I think I'd need to stare at their formula for c2 for a long time to understand where this relation comes from, though.