If each one must be assigned exclusively, and without engaging in very tempting revisionist history, i submit:
Paladin(A):Warrior(H)
Mage(A):Warlock(H)
Druid(A):Shaman(H)
Priest(A):Hunter(H)
Rogue(A)
Later:
Death Knight(H)
Monk(A)
Demon Hunter(H)
Justifications:
Paladin, Priest, Mage are literally just the names of “Human” WC3 units, and Druids are Night Elves. You cannot possibly justify them in Horde over Alliance if you are forced to choose one or the other.
Basically every single Horde unit in WC3 can be summarized as “some kind of Warrior.” Shaman is literally the name of one of their units. Warlocks were originally Orcs. Hunter is going to be a bit of a contentious pick, given night elf archers, dwarven snipers, and even the night elf “Huntress.” However, a core part of that class fantasy is the taming and use of animals, and Rexxar shows us that slice of the Hunter pie belongs firmly in camp Horde.
Rogue is our last Vanilla class, and frankly it doesn’t rightly belong to either faction. The closest Rogue adaptation in WC3 is the Draenei (that was adopted well…), but they aren’t available in Vanilla and have nothing Rogue-ish about them when they are released. I have opted to give them to Alliance in large part because it makes giving Death Knight to the Horde less of an unbalanced choice on Hypothetical Early Blizzard’s part. However, night elves having Shadowmeld lends itself to rogueish activities, and Maev is probably the closest non-draenei unit to a Rogue in WC3. Finally, when it really comes down to it, I have to ask myself: where are all the rogues coming from? They are clearly being trained, so does it make more sense for the Horde or Alliance to have institutionalized spies and assassins? Feels distinctly more Alliance to me, but this entire class is a toss-up in my mind.
Death Knights are introduced Horde for obvious reasons.
Monks are given to the Alliance. I could go on some justification about “inner peace” or “mysticism” to justify this here, but really, the astute may have noticed that in my rankings, Horde got all the mail classes and Alliance got all the leather classes. Monk is a leather class, which is by this point a core feature of the Alliance classes, and frankly, can you imagine the work to add Leather sets to the Horde drop tables? No thank you.
Demon Hunter…alright, fine, we’ll give a leather class to the Horde, stop complaining. Maybe we’ll give a Tinker class to the Alliance for mail later, or maybe they can die mad. Really though, this is about Illidan. Given the choice between the Night Elves who reviled and imprisoned him or the Blood Elves, the Horde becomes the better placement for him. It also allows him to continue his rivalry with Malfurion for future story purposes.