How can one derive the following identity, found here, relating the polylogarithm functions to Bernoulli polynomials?
$$\operatorname{Li}_n(z)+(-1)^n\operatorname{Li}_n(1/z)=-\frac{(2\pi i)^n}{n!}B_n\!\left(\frac12+\frac{\ln(-z)}{2\pi i}\right).$$
I'm curious because this formula turned out to be useful for me, but I have no idea where it comes from.
Edit: If there is a complicated proof of the above formula, but also a simpler procedure for deriving the formula for specific $n$, I would be interested in seeing both of these. I'm also only interested in the case where $z$ is real and negative.