I've tried doing the "jab his shoulder" trick, which he responds to pretty well (sometimes has an issue with going backwards instead of crossing over), but I can't get him to do it without touching him.
Let's say you were on his left side. Hold the lead like you would in show in your left hand. Have your right hand free or be holding a hoofpick.
When you ask his front to step over, start with slowly applying pressure to as far as you can reach without moving out of place. If he hasn't moved by when you are pushing kinda hard, then start wiggling the lead softly. If he still hasn't gone, wiggle it a bit more and try applying a little more pressure on the hoofpick. Keep doing this until he even shifts his weight. If he even shifts his weight, stop and drop everything down and relax every muscle in your hand. This tells the horse that what he did was right.
When you try this again, slowly
apply everything like you did to begin with. Only this time keep the pressure applied until he makes a step over. It's OK if he doesn't put his foot over the other to begin with, but just even stepping over is an awesome wind.
If he backs up, block him. For example: if I am stepping into him and he backs, I will bring me lead forward until he makes a step forward. So if he takes one step back, I will step him froward one or two steps.
My horse, he likes to step forward - so what I do when he takes a step forward I just back him up as many steps as he walked forward.
When you have to step on the other side of your horse when the judge is inspecting everything, both my horses like to pivot with me. What I do then, is when I am holding them the lead s right up by their nose but where they can see it, then when I have to step over then I drop the lead under their chin and walk right in front of their nose. -just for a little extra advice, hahaha!