That's team A in the example below. I've colored it differently since only it stays in the same position all the time. You'll see that below in the second box, where team A plays C. Use the same process to go from box 2 to box 3, and so forth until you have one fewer box than you have teams. At that point each team will have played every other team once. The following example shows the situation for 12 teams.
Any team playing team X on a given day has the day off.
No, clockwise works just as well. Take your pick, then stick with it. Any one will work. Corners are best to avoid mistakes, though. Make sure that only one team remains in the same position.