You've made a good observation. Finding what makes your pain better or worse can help the doctor or physical therapist diagnose the problem. The next step is applying the right treatment for the problem.
What you've described could be caused by one of several things. There could be a muscle imbalance in your back, trunk, buttocks, or legs. There could be a problem with the alignment of the bones in your spine, pelvis, or hips. It's possible that this movement problem doesn't have anything to do with your back pain.
The best way to find out is to have an exam. If your pain and rotation don't go away within two to four weeks, have someone take a look. Try to find an orthopedic doctor or physical therapist who specializes in back problems. Write down any other things you notice about your pain and movement. This can be very helpful for the examiner.