A migraine enlarges the blood vessels in the cerebellum. So ice helps to decrease the blood vessels. This is a medical fact.

My school nurse always hands me a ice pack to place on the back of my head says that helps with the blood flow I use ice packs a lot never tryed heat

I always put a flexible ice pack at the nape of my neck

My migraines are always on my left side too...right in the corner of my eye it feels like someone is stabbing me with an icepick through to my skull. I also have pain in my neck on the right side. So what I do is put an icepack on my eyes and a warm compress on my neck. Chiropractors suggest using ice packs and heat for muscle pains in intervals of 15-20 minutes, but the only time I used heat on my eyes was when I had sinus headaches or sinus problems. That works wonderfully!

Switch between the two in 15 min intervals.