To obtain Riemann surface for $w = f(z) = \sqrt{z}$ we get two copies of $z$-planes with cuts. After they are joined $f(z)$ gives us a one-to-one correspondence between this Riemann surface and $w$-complex plane.
Now, Riemann surface for $w = f(z) = \sqrt{z^2}$ is obtained by joining two copies of $z$-planes at point $z = 0$. What I don't understand though is how a one-to-one correspondence between this Riemann surface and $w$-plane is built. I have an intuition that in variable $w$ one, too, has to take two copies of $w$-plane and join them at point $w = 0$. After that one gets a one-to-one correspondence between Riemann surface in $z$ and Riemann surface in $w$.
Is my intuition right? Thanks.