Answer:
I look at you, you look at me, I raise my right, you raise your left.
You can tumble in it, roll in it, burn it, animal eat it. Used to cover floors, still used beyond stall doors. Freshens whatever it is placed on. Absorbs whatever is poured into it.
It stands upright and can be quite grand. Its secret is not hidden but right at hand. What is it?
What can stand in place of a country but still fold away in a drawer?
I walked and walked and at last I got it. I didn't want it. So I stopped and looked for it. When I found it, I threw it away.
This crisp stick-shaped roll is often enjoyed with soup or marinara sauce.