I explored how to make a reversible umbrella on 10 Nov and managed to make a working model for a umbrella that inverts up.
The technique to making the umbrella invert up is in the stems that pushes the umbrella open. They have to be longer than the umbrella stems at the point of contact, for the umbrella to continue to be pushed up even at the end of the stick.
The stems need to maintain their angle and cannot flatline or become parallel with umbrella frame.
If the stems are shorter or of same length as the frame at the point of contact, they will not open.
Also, if the stems become parallel with the frame, they will not invert up. The highest angle they can reach will be like a parasol, flat 0 degrees.
Therefore the stem needs to be rigid and reinforced. The frame needs to be light and rigid as well.
The umbrella frame wires should not curve, but rather be straight. Though this will reduce the curvature of the umbrella and make it more like a shallow dish than a bowl when inverted, this is better to allow a larger area of collection compared to a deep and smaller bowl shape.