Spin 1 particle through two Stern Gerlach apparatuses, one oriented along the z-axis, the other is at an angle

In class, we solved this problem:

There is a Stern Gerlach apparatus oriented along the z-axis. A spin 1 particle goes through this, the states are called $|u_+rangle$, $|u_0rangle$ and $|u_-rangle$. We let $|u_0rangle$ through a second Stern-Gerlach apparatus which is tilted at angle $varphi$ from the z axis in the x-z plane. The states after this are called $|v_+rangle$, $|v_0rangle$ and $|v_-rangle$. We know that $$langle v_+|u_0rangle = -frac{sinvarphi}{sqrt{2}}cdot e^{-itheta}$$ and $$langle v_0|u_0rangle = cos varphi.$$

The question is the argument of $langle v_-|u_0rangle$. We said that because of symmetry, the following has to be true:
$$Deltatheta=arg(langle v_+|u_0rangle)-arg(langle v_0|u_0rangle)=arg(langle v_0|u_0rangle)-arg(langle v_-|u_0rangle).$$

Therefore $arg(langle v_-|u_0rangle)=theta$.

I don’t understand why there has to be symmetry.