I'm typing based off of my knowledge of Jung's Psychological Types, MBTI and my newly (limited) acquired knowledge of socionics so keep that in mind.
Dio is some kind of aggressor, probably ILI or LIE as I think he is a Gamma and an enneagram type 8. He is the character in all of fiction that I relate to the most and see myself in. His relationship with Leda is what comes to mind when I think of Gamma duals. She has her own goals and she's using him to realize them. All Dio really wants is to prove his might, will and strength (Se Super-Id?) against Casshern who has always been one step ahead of him despite being his identical twin, but Leda slowly works him into wanting the same goals as her i.e. world domination and power. If you've watched the show then you'll know just how deep Leda's character goes.
I'm convinced that Sophita is an SEE, possibly SLE but most likely SEE. She comes across as sort of infantile at times (Ne-role?) but she's consistently an aggressor for the most part, even going as far as to outright state her interest in Casshern as soon as she met him.
Casshern and Lyuze are the two hardest for me to type and I go back and forth between a few types for each of them. I have a hunch on Casshern's type but would like to hear other thoughts.