Let $f(x) = x^{2}$, so $f(x)$ is an upward symmetric parabola. It is a perfectly symmetric function since $f(x) = f(-x)$ for any value of $x$.
Now, suppose $f$ is just some function. How would one measure how symmetric it is?
I would like to arrive at something that I could use to compare two (or more) functions and make statements such as $f$ is more symmetric than $g$.
Given any function $f:\Bbb R\to\Bbb R$, we can define two functions
$$g(x)=\frac 12(f(x)+f(-x))$$ $$h(x)=\frac 12(f(x)-f(-x))$$
Then we have $f=g+h$, $g$ is an even function (symmetric with respect to reflection in the $y$-axis), and $h$ is an odd function (symmetric with respect to a rotation of $180°$ around the origin).
If by "how symmetric" you mean with the $y$-axis, the "size" of function $h$ answers that, for however you define "size," since it is the "left-over" between $f$ and the even function $g$. If you mean with the origin, the "size" of function $g$ answers that.
There are multiple ways to define the "size" of a real function. If we allow a value of $+\infty$ for the size, we could measure $h$ with one of
$$\lim_{x\to+\infty}|h(x)|$$ $$\sup_{x\in\Bbb R}|h(x)|$$ $$\int_{-\infty}^{+\infty}|h(x)|\,dx$$ $$\int_{-\infty}^{+\infty}(h(x))^2\,dx$$
Some of these may be undefined for particular cases.
If the "size" of two functions $h_1$ and $h_2$ are both $+\infty$, we could still decide whether $h_1$ is larger than $h_2$ by comparing them with
$$\lim_{x\to+\infty}\left|\frac{h_1(x)}{h_2(x)}\right|$$
if that limit exists. There are also other possibilities, of course.
