Proof that this symmetry exists

To begin with, the electric field is defined as the negative gradient of the potential, the electric field at any point $(x, y, z)$ is $$ begin{array}{c} E_{1}=E_{0} hat{mathbf{x}}+E_{0} frac{sigma_{1}-sigma_{0}}{sigma_{1}+2 sigma_{0}} frac{R^{3}}{r^{5}}left[left(2 x^{2}-y^{2}-z^{2}right) hat{mathbf{x}}+(3 x y) hat{mathbf{y}}+(3 x z) hat{mathbf{z}}right](r>R) E_{2}=E_{0} frac{3 sigma_{0}}{sigma_{1}+2 sigma_{0}} hat{mathbf{x}}(r<R) end{array} $$

According to Gauss's Law for Electric Fields, the electric charge accumulated on the surface of the sphere can be quantified by $$ begin{array}{c} int_{V} boldsymbol{nabla} cdot mathbf{e} mathrm{d} V=int_{V} frac{rho}{varepsilon_{0}} mathrm{d} V=Q end{array} $$ and begin{equation} mathbf{e}=-nabla V end{equation} Based on Gauss's theorem, surface charge density at the interface is given by

$$ mathbf{e}_{1} cdot mathbf{n}-mathbf{e}_{2} cdot mathbf{n}=frac{rho_{s}}{varepsilon_{0}} $$ Then, the charge quantities accumulated at the surface is $$ oint_{S} rho_{s} mathrm{d} a=varepsilon_{0} oint_{S}left(mathbf{e}_{1 n}-mathbf{e}_{2 n}right)=varepsilon_{0} oint_{S} 3 mathbf{E}_{0} R^{2} frac{sigma_{1}-sigma_{0}}{sigma_{1}+2 sigma_{0}} cos theta sin theta mathrm{d} phi mathrm{d} theta $$

After a painfull of calculation, you get a symmetrical distribution.

Imagine that, before considering the charge that's on the sphere, you flip the universe in the z-direction. This leaves the sphere the same but flips the electric field because it depends on $ mathbf{hat z}$. Now flip all the postive charges with all the negative charges (this is called charge conjugation). This again flips the electric field. We are now exactly where we started. The transformation flip + charge conjugation gives you the same initial conditions so it should give you the same final charge density. This means that whatever charge density you get should remain the same after you perform the flip + charge conjugation which is the same as saying the charge distribution is symmetric.