You have an intuitive notion of the symmetry of the numbers (let's say the integers for concreteness) around $0$, and we can formalize this in mathematics. A “symmetry” is usually understood as a mapping from a set to itself which preserves some sort of structure; typically the “distance” between the elements of the set.
For example, a symmetry $f$ of the square, as normally understood, must take each point $p$ of the square to a single other point $f(p)$, and if two points $p$ and $q$ were at some distance $d$ from one another, then $f(p)$ and $f(q)$ must be at the same distance $d$ from one another. Understood in this way, the only mappings that are symmetries of the square are the ones you expect: rotations by $90$ or $180$ degrees, or reflections across one of the square's four axes.
Your intuitive idea of the symmetry of the integers around 0 corresponds to the observation that the function $f(x) = -x$ preserves distances between integers. The distance between two integers $p$ and $q$ is $|p-q|$. If $|p-q| = d$ then $|f(p) - f(q)| = |(-p) - (-q)| = |-p + q| = |q-p| = d$ also. 0 is the center because is is a “fixed point” of the symmetry, since $f(0) = -0 = 0$.
But in this sense, one could pick any integer to be the center, say 17. The symmetry around 17 corresponds to the function $f(x) = 34-x$. This mapping also preserves distances, as you can show. But instead of leaving $0$ fixed, it leaves $17$ fixed.
In this sense we can categorize the symmetries of the integers as follows:
Reflection symmetries of the type $f(x) = n-x$; these symmetries have a center at $\frac n2$. (Note that $\frac n2$ may not itself be an integer; the symmetry $f(x) = 1-x$ is a reflection around $\frac12$.) Your symmetry $f(x) = -x$ is of this type, with $n=0$.
Translation symmetries of the type $f(x) = n+x$; these symmetries correspond to sliding the entire number line in one direction or the other, and don't have a center.