Assuming each type is composed of four conscious functions/attitudes, the role of the dominant, auxiliary, and inferior functions are relatively easy to understand. The dominant is the default mindset or attitude the subject sees the world with, the filter that life is experienced through. The auxiliary is the supporting function that is the opposite of the dominant in both orientation and whether it is information gathering or decision-making. Developing proper use of the auxiliary balances out the natural dominant function, enhancing it's scope. The auxiliary is also often idealized as something to aspire to. The inferior of course is the function we are most averse to, which we avoid and which still manifests itself negatively through the type.
However, the role of the tertiary function is often obscure, or glossed over in comparison. Despite being widely accepted today as being the same orientation as the dominant function, there is controversy over whether Jung believed this, or whether all functions including the tertiary are opposite in orientation to the dominant.
Assuming the tertiary is the same orientation as the dominant, it's role in one part is played out as entering a loop with the dominant function ("Tertiary Temptation") so that the subject avoids having to use the auxiliary function to balance the dominant and so enters a loop of double extroverted or introverted functions, ignoring either the inner or outer worlds. Another related purpose of the tertiary is that it is thought to be a "comfort point" for the type, that "protects" the dominant function and the natural inclination to use it and functions of the same orientation. Or it is used/made manifest in a way other than how the function should be used, such as a manipulative tertiary Fe in young ExTPs, at least until the early twenties when it supposedly begins to be used properly in most people. Alternatively, I have also heard that it is sometimes ignored completely (unlike the inferior which still shows up negatively). So which is it? What is the true purpose of the tertiary?
As for myself as an INTP, I can relate to all of these except ignoring regarding my tertiary Si. Looking back, when growing up I can now see how I was in the Ti/Si tertiary temptation loop for many years, ignoring my Ne which appeared only infrequently, and acting very rigid and specific when it came to new experiences and certain rituals I associated with past memories, which probably influenced my confusion about whether I was a J or not (I was not "flexible" in any sense of the word, which made me think I couldn't possibly be a perceiver). I also relate to the tertiary as a comfort point. I enjoy the aspects of Si that relate to past memories and a connection to previous experiences; I can be pretty nostalgic/melancholy.
So, what is the role of your tertiary function in your experience? What do you think it's true purpose is?