I realized today that, considering the circle $ \Gamma_{\Delta} $ on the Riemann sphere whose image through the stereographic projection is the critical line $ \Delta $, the affixes of the images of its poles through the stereographic projection are the golden ratio and its conjugate, that are the roots of the equation $z^{2}-Sz+P$ with $S=1$ and $P=-1$. As those poles are antipodal points, the products of the affixes of the images is $-1$. The fact that their sum is $1$ means that one is the image of the other through the symmetry $s\mapsto 1-s$, which appears in the functional equation of $\zeta$.
RH can be reformulated as follows: there is only one circle on the Riemann sphere whose image through stereographic projection contains all the roots of the equation $\zeta(s)=0$. Hence there should be only one pair of antipodal points corresponding to the poles of such a circle.
Does this mean that the meaning of RH is that there's essentially only one possible symmetry (namely $s\mapsto 1-s$) giving rise to a functional equation allowing the analytic continuation of $\zeta$ (and other L-functions as well)?