The term "pyramid" denotes a solid with a polygonal base and triangular sides (all converging at the same point. As such is does not have to have any axis of symmetry. There are pyramids that do have an axis of symmetry and ones with planes of symmetry.
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No the white house is not symmetrical
I don't know exactly how many, but I know there are Egyptian pyramids, Aztec and Mayan pyramids, unfinished pyramids, and rock pyramids, and sand pyramids.
I believe that they were changed because of the pharoah's
The Pyramids of Giza
Ziggurat