Palpitations are a common complaint amongst those with mitral valve prolapse. Without having more information it is difficult to discern exactly what the rhythm is when you have these palpitations. Palpitations are generally benign when short lived. If you have prolonged symptoms or the symptoms change, you and your doctor could consider an ECG monitor to make sure you don't have a more serious arrhythmia.
I am not aware of any anatomical explanation to explain why these happen when you lie on one side and not the other. It is possible that when you are lying on your left side, the heart swings closer to the chest wall and you may be more aware of the skipped beats because you can actually feel them.
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