If they weren't crossed you would have a null straight down the axis.
Picture one side making current go outward from the center.
If you have the other side be the same, electrons move from the center outward on both arms.
You want the currents on both sides going in the same direction so they add up down the axis. Hence one side has to be fed 180 degrees out of phase from the other. That makes both sides go to the left at the same time, then they reverse and both go to the right.
Similar to making two identical dipoles with ground plane. planes, locate one one atop another, one points up, one points down. Feed them with a 0/180 hybrid. The zero port (same phase) looks like a null on the horizon, 180 port peaks on the horizon. Same idea for the log periodic.