If $p(x) = \sum_{j=0}^n a_j x^j$ is a polynomial of degree $n$, and $r_j$ are its roots, there are usually not very explicit relationships between transformations of the coefficients $a_j$ and the roots $r_j$. One very cute exception is the map $$ \sum_{j=0}^n a_j x^j \mapsto \sum_{j=0}^n a_{n-j} x^j $$ which transforms the roots as $r_j \mapsto 1/r_j$ when $r_j \not= 0$, and converts a root $0$ of multiplicity $k$ to a root $0$ of multiplicity $n-k$.
The proof of this is to move to a field where $p$ splits into linear factors, notice that the transformation is also given by the formal multiplication $p(x) \mapsto x^n p(1/x)$, and inspect what that does to the factorization of $p$ into linear factors.
This is certainly very classical, but does anyone know a reference for this fact or, more interestingly, some applications of it?