Independent Brownian arms at a deterministic time

For a deterministic $h>0$, the processes $B_{h-t}-B_h$, $0\le t\le h$, and $B_{h+t}-B_h$, $t\ge 0$, are independent standard Brownian motions. The first uses reversed, sign-changed past increments; the second uses future increments independent of the entire past. The first process is not generally independent of $B_h$, but translating a range by $B_h$ preserves its Lebesgue measure pathwise.