Nontrivial Symmetry of an Integral Depending on a Parameter

Question

We have defined the following function \begin{equation} \chi(\gamma)=\frac{1}{2\pi}\int{\rm d}^2{\bf x}_2\frac{{\bf x}_{10}^2}{{\bf x}_{20}^2{\bf x}_{21}^2}\left[\left(\frac{{\bf x}_{21}^2}{{\bf x}_{10}^2}\right)^\gamma+\left(\frac{{\bf x}_{20}^2}{{\bf x}_{10}^2}\right)^\gamma-1\right] \end{equation} In this expression ${\bf x}_i,~i=0,1,2$ are two-dimensional vectors and ${\bf x}_{ij}\equiv {\bf x}_i-{\bf x}_j$. We want to show that $\chi(\gamma)=\chi(1-\gamma)$ without explicitly solving the integral, just by inspection of the symmetries of the integrand. We suspect that this property has to do with symmetry under inversion ${\bf x}_i\to {\bf x}_i/{\bf x}_i^2$. Notice that the three terms in the bracket must be taken together in order to have a finite result as ${\bf x}_2\to \infty,~{\bf x}_2\to{\bf x}_1$ and ${\bf x}_2\to{\bf x}_0$.