Given a function $f$ and its Fourier transform $\hat{f}$, then the function
$$F=f^2+\hat{f}\star\hat{f},$$
with $\star$ the convolution, is its own Fourier transform. So I would say that the question "are all the distributions $T$ such that ${\cal F}T=T$ known?" has the answer "no, there is an infinite number of these".