So I think I've figured out how Socionics and Enneagram are related. Here's my theory.
As described in this article, our psyches are composed of our Nature and our Persona. Nature being the immutable core of who we are, and Persona being the image we project to function in the world. Sociotype is an aspect of our Nature, and I believe the evidence indicates that Enneatype is an aspect of our Persona.
The reason for this is that, as described in the Enneatype Overviews listed on EnneagramInstitute.com*, Enneagram Types develop primarily as a result of one's relationship with one's parents. The first obvious correlation between Sociotype and Enneatype is that one's Sociotype, along with the Sociotypes of one's parents, will play a large part in determining the nature of one's relationship with one's parents. A person with a Duality relationship with both parents imay well become a Nine, barring extenuating circumstances, whereas a person with a Conflict relationship with both parents may be more likely to become a Four, barring extenuating circumstances.
However, I think the biggest link between Socionics and Enneagram is via the DCNH subtype system. Like Enneatype, in my opinion subtype is an aspect of the Persona, not the Nature (I'm sure JohnDo would disagree with me on that, but his conception of subtypes is misguided). Because they both belonged to the Persona, I instinctively felt that there must be some relationship between Enneatype and Socionics subtype, but I was looking at it backwards: I thought that one's subtype must determine one's Enneatype. I am now convinced that it is the other way around: one's relationship with one's parents determines one's Enneatype, and one's Enneatype determines one's subtype.
Essentially, Enneagram type determines the role one plays in one's family. Each role calls for a certain type of behaviour, which will require the use of certain information elements that may or may not be strong in one's Sociotype. Repeated use of these information elements results in a DCNH subtype.
For example: an LII 5 grows up feeling ambivalent about both his parents: he's not getting all the support and guidance he really wants from them, but he still feels that if he can figure out a role to play, he could win his parents' attention. However, no role to play is immediately apparent, so he withdraws to try to figure something out. They try to figure out the world, in order to figure out their place in it. This systematizing task requires a focus on Ti, leading to a Ti-Normalizing subtype.
On the other hand, an LII 4 grows up feeling entirely disconnected from both parents, and so gives up entirely on ever getting any support and guidance from his parents. Lacking role models, he therefore turns inward to his imagination in order to find information about himself. This task would require a focus on Ni, leading to an Ni-Harmonizing subtype.
Anyway, that's my theory. I haven't figured out all the details yet -- it does seem that types 4 and 9 correlate with the Harmonizing subtype, and 5 with Normalizing. How does all this match up with everyone else's experiences?
*Enneatype Overviews can be found at these links:
1—Enneagram Type One: The Reformer—Overview
2—Enneagram Type Two: The Helper—Overview
3—Enneagram Type Three: The Achiever—Overview
4—Enneagram Type Four: The Individualist—Overview
5—Enneagram Type Five: The Investigator—Overview
6—Enneagram Type Six: The Loyalist—Overview
7—Enneagram Type Seven: The Enthusiast—Overview
8—Enneagram Type Eight: The Challenger—Overview
9—Enneagram Type Nine: The Peacemaker-Overview