A classy unexpected move
1° Pull on the bars for a backhop, and give a strong twist like trying to look back.
2° In the air, follow up with your momentum and realigne the bike with your shoulders.
3° The move continues, past the realigning of the bike with your body to a complete turn.
First, align yourself as close as you can to the side you want to jump. The tyres should be virtually on the edge, or standing on the last three inches of border. Land with the back wheel first, that lets you more flexibility for the finish. See this move in a video
Turning with a kick
Another way to launch these sweet 360° drop off is to give a pedal kick while throwing yourself on the same side as your driving pedal. That seems a bit odd at first, but the pedal kick gives you extra power while you start the motion with shoulder movement and a good twist in the handle bars.