Where empirical methods and subjective methods cross, existentialism and psychoanalysis result. Should existentialism be here? I don't like, first of all, that existentialism is organized with psychoanalysis. More importantly, it is hard for me to believe that existential psychotherapy is easier to research than Rogers' common factors (as Yalom's book on group psychotherapy can attest). That being said, I think that Casile, being an existentialist himself, knows better than I on this one. This area's title is Rationalism. This could possibly be because these two orientations are the deepest, most roots-based approaches. But rational? The use of psychoanalytic metaphor is one of the less rational approaches I know (not necessarily saying it's false - metaphor can be a powerful thing).
Humanism is the convergence of experiential and subjective. Humanism is all about the client. But not only. Humanism is also all about the therapist. While growth and personal reality are important in the client's life, I think that it is just as important to the therapist. I think that the best and perhaps most vocal of experiential proponents was Carl Whitaker who always wanted the client to act as his therapist. I think that everything in the Humanism category is also more feelings-based than its polar opposite, Empiricism. Experiential models are inherently more about affect anyway.
Empiricism is more the polar opposite of Humanism. It does not dwell on emotions, but rather on behaviors or cognitions. Both behaviorism and cognitive-behavioral therapy are testaments to that. They are both the most empirical theories to date. They also espouse a relationship that is one of teacher and student. The client is actually being taught how to cope better or become more active in their own lives. This teaching underscores the "correctness of reality" that its place on the x axis denotes.
I can't say that I know much about the last category. My program does not intensely teach the more collectivist theories (outside systems-based family approaches). I honestly don't think that systems theories should be in this model at all. It seems to me that systems and multicultural competencies are better used as a z axis. What to call such as axis? At this point, I do not know. Perhaps later I will find a phrase that rolls off the tongue eloquently enough to sit somewhere beside such an educational model.