You definitely don't need a round pen. I actually find them somewhat restrictive, unless I'm doing liberty work--in which case you need at least some kind of fencing.
I ran into a problem getting my horse to circle to the right this year. I don't work on a lunge line, just a 12-foot lead (on occasion, a 22-ft). Her right side has always been her tougher side, but late this summer, she started backing and turning when I asked her to circle to the right. There were a few issues at play, but most importantly, we determined that she'd developed a mental block with going to the right because that's the side her saddle had started pinching on. She began to associate going to the right with pain in her shoulder. (I have since bought her a new saddle that fits perfectly, and we will likely have to spend some time working some bad habits out. Unfortunately, I live 6 hours away from my horse so probably won't see her again for a few weeks!)
Anyway, I have managed to get her circling to the right again with more ease, though there is still some work to do. Here's what we did:
1. To the left, I simply have to stand parallel with her at the drive line, and then just send her on with my body language--pointing with my left arm, and looking where I want her to go, not at her.
2. When I wanted her to go to the right, it was a struggle just for me to get to her right side. She wanted to back and turn so I couldn't get to that side at all.
3. I worked at putting her to work whenever she wouldn't stand for me to go to her right side. When I could finally stand in close to the drive line on her right side, I would allow her to rest, and rubbed her everywhere to put her at ease.
4. When I could get to the right side without a battle, and could stand in close while she relaxed, we started to work on forward motion. I would send her out with my right arm (pointing the direction I wanted her to go), just like when we circle to the left, BUT...I would stay RIGHT beside her, touching her body at the drive line, and walk forward with her, in an arc. This kept me close enough that she couldn't back away from me or turn--I could go with her wherever she went--and helped her understand what I was asking--I wanted forward motion.
5. Each time I did this, the forward motion became easier, until I could step back from her, and begin to circle her from more of a distance. If she would revert to wanting to back and turn, I'd go back to walking in close beside her until we worked it out again.
6. When I could get her circling at a distance, I started *small.* If she gave me even a quarter of a good circle with forward motion, to the right, on an arc, I'd ask her in and let her rest. I'd then work on increasing the circle size.
When I last saw her, we'd progressed to where I could ask her for a right circle from a few feet away from her, saddled or unsaddled, and she would respond without problem. When I see her again in a few weeks (at the time I'd been home for 3 weeks and had been working with her regularly), I'll likely start in close to her, so we can start on a positive note, and work the distance out quickly if she's responding well.
Patience, and quality will get you everywhere. Good luck!
P.S. It's also a very good idea to get your horse checked by a chiropractor and/or massage therapist. It's not impossible that there's some physical discomfort influencing your horse's behavior as well.