I'm looking for an algebraic identity (if it exists) that relates $\text{Li}_n(-z)$, $\text{Li}_n\left(\frac{1}{1+z}\right)$ and/or $\text{Li}_n\left(\frac{1}{z}\right)$ for $z > 0$ and $n \in {\mathbb N}$. The identity should include only polylogarithms involving these arguments (but could mix the orders, if needed).
I've searched far and wide and have come up with nothing. The only thing I could find is a identity for the dilogarithm $$ \text{Li}_2 \left( \frac{1}{1+z} \right) = \text{Li}_2(-z) + \log z \log(1+z) - \frac{1}{2} \log^2(1+z) + \frac{\pi^2}{6} $$ Perhaps such an identity does not exist, but I thought I'd confirm it here first.
PS - If you're wondering why I'm looking for such an identity - I have a function (which solves a differential equation of interest to me) which is a complicated mess of polylogarithms with arguments above and I'm hoping to consolidate and simplify the result using as many identities as possible.