The question is somehow similar to Can I use LGA775 heatsink on LGA1156 CPU? although I didn't quite get the answer and the point might be a bit different. I don't have the stock cooler for the 1155 ...
Back in the day I used a desktop with Celeron D 3.2GHz on LGA 775 mainboard. It came with some stock radiator and cooler which worked OK. Later I upgraded the rig with Pentium 4 2.93 Prescott (non-HT) ...