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The diagonals of rectangles are rotational lines of symmetry but not reflective. To be reflective lines, folding along the line has to give the same shape on each side.
Six - three from diagonal to diagonal, three from midpoint of a side to midpoint to a side.
It depends on the nature of the irregularities. If you have an octagon where each alternate side is equal then you can have 8 lines of symmetry.
3 Lines of symmetry
A parallelogram need have no lines of symmetry.