Pammy, you can stand on your head to get the correct line if that's what it takes to get it right.

The purpose of symmetry, in what I teach anyway, is to provide the mind/eye/hand skills to produce an object that is symmetrical. Yes, that sounds obvious, but think of what else has symmetry - that cup you're drawing, a vase, a ball, a face. By splitting your image in half, and producing one side first, then the other, you'll ensure that your drawing is balanced.
The other reason for doing it this way is to encourage the right side of the brain to take over. The left side is analytical and wants to label things, this exercise is abstract shapes and doesn't let the left side of your brain label them because it doesn't know what they are. So the right side, the creative side, takes over and looks at shape and tone.
So how to create those shapes? Relax and concentrate on your task. There is no law that says you have to have your paper a certain way to achieve the results that you want. Do whatever is comfortable to get the shapes that you want. In complex shapes such as these, you may find it difficult to create those shapes without turning your paper, but try it without turning. It will feel awkward and difficult at first, that's the left side of your brain telling you that it's not right. Ignore that and keep going, soon your right side will take over and let you finish the task.