I've been looking into these recently, since they're pretty interesting. Basically the HLA genes are more actively evolving than other genes, hence it might be a place to look at regarding "new" diseases. It's pretty wild, since they've got missense mutations all over the place, and it's quite common to see SNPs there with 3 or 4 possible alleles, instead of the usual 2 (or 1).
From what I can see, HLA phenotypes are completely determined by SNPs. The primary ones looked at for disease risk and transplant rejections are DRB1 + DQA1 + DQB1. Though I've also seen phenotypes listed for DRA, and was able to determine full DRA phenotypes for 6 ME patients based on their 23andMe data, and narrow it down to three phenotypes for the remaining 5 ME patients, of which each would have two different ones:
Patients A & B: DRA*010202 (homozygous)
Patient C: DRA*010101 (homozygous)
Patients D, E, & F: DRA*010101 + DRA*010203 (heterozygous)
Patients G, H, I, J, & K: Two of DRA*010101 or DRA*010201 or DRA*010202 (heterozygous)
Unfortunately DRB1 is exponentially more complex, and the 23andMe testing on the gene is extremely superficial - just 5 or 6 out of 200+ relevant SNPs are tested. And the ones that are tested aren't the useful ones! But it's possible that some of the 23andMe SNPs near the gene will indicate which alleles we have in the relevant SNPs. DQA1 and DQB1 look similarly complex and sparsely tested.