Would spuds grow? I went to Potato Day at Garden Organic one year, and remember being told that potatoes actually don't need anything to grow in, there's enough sustenance in the tuber to grow a crop, albeit a smaller one than if the spuds are planted into something more nutritious.
If you pushed spuds into the hay and let them do their work, you'd get some spuds and also quite a lot of reorganisation / composting of the hay, wouldn't you? People used to do it on waste / used straw, I can't see why it wouldn't work on hay.