Let $a\in \mathscr S'(\mathbb R^{2n})$: you can quantize that tempered distribution, i.e. associate linearly to $a$ a linear operator $\text{Op} a$
from
$\mathscr S(\mathbb R^{n})$ into $\mathscr S'(\mathbb R^{n})$ with the formula,
where $u,v\in \mathscr S(\mathbb R^{n})$,
$$
\langle (\text{Op} a) u,v\rangle_{\mathscr S'(\mathbb R^{n}), \mathscr S(\mathbb R^{n})}=\langle a,\mathcal H(u,v)\rangle_{\mathscr S'(\mathbb R^{2n}), \mathscr S(\mathbb R^{2n})},
\tag{$\sharp$}$$
where the Wigner function $\mathcal H(u,v)$ is defined on $\mathbb R^{2n}$
by
$$
\mathcal H(u,v)(x,\xi)=\int e^{-2iπ z \xi} u(x+\frac{z}{2})
\bar v(x-\frac{z}{2}) dz.
$$
It is easy to prove that $\mathcal H(u,v)$ belongs to $S(\mathbb R^{2n})$
when $u,v$ are in $S(\mathbb R^{n})$ and this gives a meaning to $(\sharp)$.
Now when you have the symbol, and if you use the classical quantization formula, you get the kernel by the formula
$$
k(x,y)=F_2a(x, y-x),
$$
where $F_2$ is the Fourier transformation with respect to the second variable, which makes sense on $S'(\mathbb R^{2n})$. This also implies that the symbol $a$ can be expressed in terms of the kernel with

$$
a(x,\xi)=\int F_2a(x, z) e^{2iπ z \xi} d\xi=\int k(x, x+z) e^{2iπ z \xi} d\xi,
$$
where the integrals should be understood in a weak sense on $S'(\mathbb R^{2n})$.
As a result, that reversible relationship between kernel and symbol always holds.
Symbols are in general nicer to manipulate than kernels: the kernel of the identity is $\delta_0(x-y)$ whereas its symbol is 1. The Hilbert transform has the kernel ($y,x\in \mathbb R$)
$$
\frac{1}{iπ(y-x)}
$$
i.e. is the convolution with $pv(i/π x)$ and has symbol $\text{sign} (\xi)$, and for instance the $L^2$ boundedness of that operator is trivial knowing the symbol since we have a Fourier multiplier by a bounded function,
whereas that property is not obvious when you look at the kernel. A parametrix of the Laplace operator in $d\ge 3$ dimensions has kernel
$$
c_d\vert x-y\vert^{2-d},
$$
a locally integrable function which is also a temperate distribution,
and its symbol is $\vert \xi\vert^{-2}$.