Considering the diagram attached, we're mainly concerned with convex quadrilaterals. You can examine each of the various types to see which have rotational symmetry.
The parallelogram is the top of a chain of quadrilaterals that exhibit 180° rotational symmetry. The ones connected by lines below parallelogram in the attached diagram are the same as the ones connected by the lines in your diagram above.
Perhaps you can identify in your mind's eye what the lines of symmetry are in each case. If not, perhaps it would help to draw the shape on a piece of paper and rotate it to see what rotation makes it look the same. Likewise, you can fold it on the line of symmetry and see where it matches with itself.
The appropriate fill-ins are
... ... parallelogram
rhombus ... rectangle
... ... square