Why do spring onions curl up when placed in cold water?

This happens because the inside of the stem soaks up water, increasing its volume. The outside cannot soak up water, and stays at the same volume. This is pure physics/geometry and happens in many other occasions - dandelions come to mind when thinking of plants, but the same happens in stockinette knitting. It is because it is a shape similar to a flat sheet, but one side of the sheet is larger than the other one and tries to "expand".

The relationship to "how is cut" is minimal. Sure, if you cut it into a different shape, the effect may be so small that you won't notice it. For example, if you cut the onion stems into ringlets, the pull will be symmetrical in all directions and the outer layer will be too stiff to allow even the two cut ends to curl. If you cut them into small squares instead of long strips, you will get less curl and more of a saddle shape - but it is not that something different is happening, the same things are happening in all ways of cutting, and the geometry of it determines which effect will dominate the others.