Mine doesnt like right either, Its a common problem caused by us.
We mounbt on the left, lead from the left, interact from the left. Horses brains arnt connected like ours, It is almost like they have two brains a right and a left. Frequently we never train the right brain. Also we lunge and just go round and round to the left because it is easier. You should work on what is hard. My horse originally wouldnt lunge either direction and was afraid of carriage sticks or lead lines sometimes he would strike out with his front fee. Eventually I got him moving and going where I point. I stand with lead line in my left hand about 4 feet of slack, raise my left hand and point. I then tap him on the withers till he goes, each tap gettingharder. Then repeat going to the right. Admittedly I had to get on him a bit to get him moving to the right but once he figured it out he was fine. Now he moves when I just point lead line or not. Going left still isnt perfect though.
For about a month start handling him on his right side, every thing from feeding to grooming, lead him from the right, put his saddle on from the right, mount from the right, and MAKE him go where you point either left or right.