Let $M$ be a matroid and $T(x,y)$ its Tutte polynomial. It is well-known that $T(y, x)$ is equal to the Tutte polynomial of the dual of $M$. What kind of matroidal information (if any) is contained in $\frac{T(x, y)}{T(y,x)}$?
One can assume that either one can evaluate $\frac{T(x, y)}{T(y,x)}$ at certain points, that one has access to the coefficients of the expansion of $\frac{T(x, f(x))}{T(f(x),x)}$ for some simple polynomial function $f(x)$, or some other reasonable notion of access to $\frac{T(x, y)}{T(y,x)}$.