Ok, after posting this on the Socionics.com forum, looking at the socionics.us rat/irr descriptions, and some further consideration of the Reinin Dichotomies, I've begun entertaining the possibility of being ILI/INTp. Based purely on dichotomies, I would appear to be ILI. I can certainly see the ILI profile descriptions listed here and on socionics.com fitting me. I've considered my relationships with people I've typed, and the intertype relationships seem to fit with me being ILI.
So, can anyone else share some differences between ILI and LII to help me further evaluate this possibility?