You got some great answers form folks. The one I cued into the most was the person who said that his feet are stuck. I think that point is very important. When a horse gets stuck like that , you have to break him out of it, like breaking out of being frozen to ice.
The one person who suggested unbalancing him was right in that this makes him put more weight on one side than the other. The same thing can happen if you ask for lateral movement .
Do you know how to "disengage" his hindquarters? If you are trying to back him up straight and he gets stuck (and He isn't smart or evil enough to intentionally try to outdo you in stubborness), you will have to give up on the straight backup and get him loose. (btw, does he feel stiff when he is backing up?) You will have to put a bend in him and may have to go so far as to disengage his hindquarter to break him out and then start again.
Remember to kind of lift your pelvis just a tiny bit off the saddle and believe it or not, you kind of put your upper body the tiniest amount forward and ask the horse to back into the space you have provided for him. If you are leaning too far back with a hard seat, digging your seat heavily into the saddle, you are anchoring him to the ground. Actually think backup in your head. Visualize how it feels and which foot moves next.
When you apply the rein, apply one more than the other, should create a give in the jaw. He should back one step, on that side. You do the same on the other side.. Step by step with a bend on each side.
Oh, I just thought of somethig. If you are riding in a curb or Tom Thumb, it won't work as well as if you are working in a snaffle.
Anyway, if the horse doesn't listen to the bend/back cue of the rein, then take it back so far that you actually disengage the hindquarter on that side, and the horse steps under. Why do that? Because it breaks him out of the resistance. Then without getting angry, try backing up again. Pretty soon he will find that backing with a little bend in his body is easier than doing the whole disengagement. He will become softer, too.