Lets talk first talk haemoglobin. Our haemoglobin picks up oxygen in the lungs and drops it off at the tissues in a very nice, elegant system. But for us, as the temperature of our haemoglobin gets lower, it holds onto the oxygen, more and more tightly. Not a problem for us, as we don't get very cold (thank you homeostasis). But, if this was the case for our reindeer - that would be a real problem - as legs are probably one of the main places it needs oxygen for doing such things as maybe running away from a wolf or two. Luckily reindeer haemoglobin is different, not only do cold temperatures not prevent the dropping of oxygen - it may actually encourage it! Way to go evolution!
Now lets get back to reindeers.
So unlike most organisms, reindeer have different chemical compositions of their cell membranes depending on where they are in their body - more saturated fatty acids at the rop of their legs and in their core, and less and less as you go down their legs.
Which is (at least to my mind) enjoyably clever for Santa's little helpers!