We have learned that any "strictly positive monotonous transformation" of utility functions is okay, as long as they preserve the ranking of choices implied by the underlying preferences.
Consider $U(x, y) = (x/y)^\alpha$
The cross derivative is $\frac{\partial^2}{\partial y \partial x} U(x,y) = -\alpha^2 (\frac{x}{y})^{\alpha - 1}$
Now, consider the monotonous transformation $V(x,y) = \log U(x,y) = \alpha \log x - \alpha \log y$. The cross-derivative of $V$ is $0$.
The cross-derivative incorporates important information on how the ranking of choices of $x$ changes, as we change $y$. Clearly, these two different cross-derivatives cannot be generated from the same underlying preference ranking - or am I mistaken?