As I’m lying down on my porch swing, I find it trivially move the swing parallel to the way that I’m lying by merely moving a body part back and forth in the same direction. EG: Raising and lowering an arm. I assume that this movement is caused by shifting my center of gravity. However, no combination of movements that I can come up with will cause the swing to move perpendicular to my body (the way a swing traditionally moves). If it is a changing center of gravity that causes the parallel movement, why doesn’t side to side movement of my arms or legs produce the same result?
And yes, I realize that I could just use my feet and legs. 
My thoughts:
I assume the porch swing is hanging from two points, one at each end.
Now if you’re lying the long way, your center of gravity is going to be around your navel, in the center of the swing. So your arms start out a foot or more from the center of gravity in the parallel direction, but only a couple inches in the perpendicular. So it’s easier to shift your center of gravity in the parallel direction.
Also, when you do shift in the parallel direction, you’re going to shift it directly in the parallel direction, staying in line between the two suspension points, so the resulting movement is going to be in a stable swinging pattern. But when you try and move in the perpendicular, it’s going to be unbalanced from side to side, so the swing twists a little bit. That’s going to make the swing twist back as it recovers, and likely cause most of the motion to be dissapated rather than going into a stable swing.
Yes, you are correct about how the swing is hanging. I’ll try experimenting with waving a 2x4 near my navel and see if the results change.
Your body is also more stable lengthwise than side-to-side. When you move your arm head-to-toe, your body and the swing will tend to move together as a unit. When you move an arm left-to-right, you may just roll from shoulder to shoulder rather than causing the whole swing to move with you.
So, for that 2x4 experiment, have someone strap you down, tightly.
I don’t know why you can’t, but I can. It’s a little harder than the lengthwise motion, but that might just be because it’s easier to move one’s center of mass further lengthwise.