Does negative zero exist? In the set of real numbers, there is no negative zero. However, can you please verify if and why this is so? Is zero inherently "neutral"? 

A common definition of negative is "less than zero". In this sense, zero isn't negative (nor positive for a similar reason).
But the opposite of zero is well defined: it is zero. The unary $+$ or $-$ operators can very well be applied to $0$, with no effect.
One can admit that zero has no sign. The $\text{sign}$ function is usually defined to be $-1,0$ or $1$.
