I made a thread on this topic a while ago, but I wanted to elaborate on the distinct roles of each complimentary functional pairing to develop a clearer sense of each function's role in the larger picture of quadra progression.
Delta is the quadra of establishment and unity. Under Delta, all aspects of life are organized and unified into a sustainable whole that attempts to treat each individual according to their needs and maximize their usefulness for the perceived common good.
The Te/Fi axis of Delta is primarily concerned with the implementation and standardization of productive and useful methods and the spreading of universal morality. They seek to establish holistic goals, rather than specific ones, that are geared towards universal applicability and sustaining an established implicit communitarian structure. Their goal is to foster sustainable productivity and establish universal ethical standards that everyone can live by.
The Ne/Si axis of delta is primarily concerned with the continual refinement of established structures and ethical principles. They like to make "little changes here and there," constantly perfecting and refining various standardized procedures. Their goal is to maximize their usefulness and that of those around them by continual maintenance of their lifestyles. They are sincerely open and questioning regarding matters of morals and ethics, maintaining a generalized standpoint with common sense ideas of right and wrong while adjusting their own perspective to align with that of a sense of common good.
Alpha is the quadra of new beginnings. Alphas seek to foster creative activity and promote holistic change in their environments and people's ideas.
The Ne/Si axis of Alpha is primarily concerned with generation of ideas and maintenance of environments conducive to fun, intellectual stimulation, and creativity. They examine and challenge existing standards on a purely intellectual level, rarely seeking to dogmatically criticize or berate, but rather to break things up piece by piece and attempt to put them back together in the proper order. They are primarily focused on ideas as intellectual "toys" that may or may not develop into something worthy of support or pursuit; when they are not, they are used for their intrinsic worth and abandoned when they prove useless. When the Ne/Si axis of Alpha unearths ideas worthy of pursuit, it seeks to spread them in a grassroots fashion, appealing to those whom they think will be interested in mutual pursuits.
The Fe/Ti axis of Alpha is primarily concerned with the structuring and implementation of ideas that are seen as beneficial into everyday life. They often organize into loose collectivities that can resemble the tendencies of Aristocratic quadras, but are actually sublimations of common goals and a mutually perceived broader need for a sense of unity and collectivism.
Beta is the quadra of rebellion and disruption. Betas seek to organize in the name of uniting under a worthy cause in order to enact drastic change, and to do whatever is necessary to accomplish their ends.
The Fe/Ti axis of Beta is primarily concerned with the organization and standardization of ideas in order to unite. They are wary of doubters or non-believers, wanting everyone in their Aristocracy to agree on the matters that are deemed most important so that they can be united in common purpose and with equal resolve. Their role is to garner support for their cause and develop a central theme for organization that will carry them through to the end. Their purpose is to build the platform from which a final plan can be launched.
The Se/Ni axis of Beta is primarily concerned with making the necessary push to accomplish their collective goals. Their focus is on determining when and where to strike for optimal impact. Typical tactics include inducing chaos intentionally, such that, though anyone affected, including themselves, is at a disadvantage for lack of structure or certainty, at least they are ready for it, and as such can capitalize on the lack of organization in order to establish a position or make a vital move. Their role in quadra progression is to do that which others are not willing to do in order to enact necessary change.
Gamma is the quadra of reform, seeking to create stability in their environment, cultivate their individual potentialities, and organize themselves and others in pursuit of broad-scope productivity.
The Se/Ni axis of Gamma is primarily concerned with steadfastness and genuine expression. They are acutely aware of the consequences of their actions, both concrete and personal, and are wary of those who seem blindly devoted or pious. They seek to surround themselves with others whom they can trust, searching out companions with strong character and moral fiber and banding together to "weather the storm." They are guided by common sense morals and a passionate sense of knowing their loved ones fully and being there for them at all costs under any circumstances. Their role in quadra progression is to jolt others back to reality, to properly assess their circumstances, and take the necessary action to begin
The Te/Fi axis of Gamma is primarily concerned with organization and implementation of effective methods. They often live by an explicit personal code that is rooted in their own self-interest and that of those close to them. Perhaps more than any other pair, they pursue development of their own personal latent positive qualities as directly and fully as possibly in order to maximize their strength and usefulness. Their role in quadra progression is to promote personal growth and begin the natural process of organization and standardization that lays the ground work for collectivization.
This is a work in progress; feel free to criticize and contribute if you take exception to anything I've written, think I'm missing something, or think I could elaborate more on a specific point.