Is there a symmetry argument that the running minimum of a Wiener process is the negative of the running maximum of that process?

Question

Wikipedia provides a derivation of the running maximum of a Wiener process, which turns out to be a half-normal distribution. The resulting expected value of the running maximum of the Wiener process is simply $\sqrt{\frac{2t}{\pi}}$ where $t$ is the time. Would the running minimum also have a half-normal distribution, resulting in an expected running minima of $-\sqrt{\frac{2t}{\pi}}$ via an argument of symmetry?