As a thought for discussion, why do you think the frequency domain mirrored across the center point?

The frequency domain can be though of as a map of exponentials, and $\sin(2\pi x/16)$ can be rewritten as $-\frac{j}{2}e^{2\pi j x / 16} + \frac{j}{2} e^{-2 \pi j x / 16}$

More genrally, all real-vaued signal should be mirrored. In frequency domain we convert signals into exponent base of $e$. For real-valued signal, the imaginary part of the expression has to be eliminated, so it is mirrored acrossed the center point.