To keep the rings round, you will need to twist the ends open; never simply pull them apart. If you do pull the ends away from each other, it is very difficult, if not impossible, to get the ring to lie flat.
You will need to hold the ring with the cut ends in the 12 o'clock position and have the ring firmly grasped on either side. Try and cover as much of the ring with your pliers as possible to avoid twisting the ring out of shape. One end of the ring is pulled toward you and one end is pushed away.
When closing a ring what you need to do is to create tension in the metal to push the two ends of the ring against each other making sure you get the smoothest, closest join you can.
You are going to twist the ring closed reversing the action you used to open the ring. As you twist the ring closed, you will need to press in slightly so that the ends of the rings slightly overlap. You then will use a small amount of pressure to slightly open them and the ends pass each other and slip into place. You'll hear and normally feel the ends rub against each other as they slide into place.
The tension you create in the metal by pushing the ends past one another will keep them pressed tightly together when they are closed. If the ends do not line up perfectly the first time, simply repeat the procedure in smaller increments until they do. Please be aware though that you can overwork the metal and make it brittle if you do these steps too often.