Okay. Well I was pondering some more on the quadras today and their social roles instead of just thinking about behavioral characteristics and here's what I came up with:

Alpha: conservative
Beta: reformative
Gamma: reformative
Delta: conservative

It's seems odd to call the Alpha quadra "conservative," especially with those ENTps in there, but then again, while the NTs come up with brilliant ideas (What should we be achieving?), Beta actually tries to implement the ideas into reality (How should we be achieving it?).

Also, it seems that Gammas and Deltas are more socially-oriented while Alphas and Betas are more ego-asserting.

So to tie this all together:

Alphas are ego-asserting conservatives who question what they should be achieving.
Betas are ego-asserting reformists who question how they should be achieving things.
Gammas are socially-oriented reformists who question what we should be achieving.
Deltas are socially-oriented conservatives who question how we should be achieving things.

Does this make sense?