To explore symmetry with my preschooler, we got out the paints and paper. My 4 1/2 year old grabbed the paint tube to use the lid as a stamp, placing color on one side of the paper. Then he folded the paper in half, and pressed down on it.
I asked him what he thought would happen when he opened the paper. He was so excited to see what happened as he pulled open his painting.
He laughed as he saw how the colors and shapes matched on either side of the fold. The next time he added a burst of color to one side of the page, I asked him where he thought the color would go after we folded it. He pointed (approximately) to the corresponding spot on the other fold of the paper.
He kept adding dots of color, and refolding. He had so much fun with this activity that I ran out of room to hang up all the sheets to dry!