Does Cauchy Principal Value always exist?

Question

Is there any theorem that guarantees the existence of Cauchy Principal Value? I know that for many improper integrals, the Cauchy Principal Value exists but the improper integral might not exist, such as $$\int_{-\infty}^{\infty}x\,\mathrm{d}x.$$ So, I am wondering the reason that leads to such a result. Some says that it lies under the fact whether the limiting process is symmetric or non-symmetric. This seems to make sense. But why? Is there any more detailed explanation? Thank you!