Oh, dude, you're asking about a kite! Yeah, a kite doesn't have rotational symmetry and its diagonals are not perpendicular. It's like that one shape that's just doing its own thing, not conforming to the norms of the quadrilateral world.
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A parallelogram
cone
It is a kite.
Square, circle, equilateral triange! Hope I helped! :)
parallelogram * * * * * A parallelogram does have rotational symmetry (order 2).