What do the material and the spiritual have in common? I know that spiritual forces and beings guide all material matters, and for that to be so, there must be a certain quality that both sides share. I’m asking what precisely that quality is?

To say that we can we can point to a particular physical quality which connects the spiritual to the physical seems to be oversimplifying. That would mean that someone could take, let say vitamin "S" and then he would be more spiritual. I believe we realize that things don't work like that. Being more spiritual is by fulfilling the will of Hashem. One of Hashem's wonders is the very fact that he connects the spiritual to the physical. This is something we bless Hashem for, every time we say the blessing of "Asher Yatzar" which ends with the words "Mafli La'asot= He does wonders". The Rema in the Shulchan Aruch 6:1 explains that the wonder is that Hashem preserves the spirit in man and connects the spiritual to the physical. The Kuzari (Part I: 99, Part II: 27) says though the soul dwells in the human body it is there as a leading and guiding force not that it actually has a presence in the physical realm since it does not benefit from the food of the body. It is way beyond that. To explain this idea, the Kuzari quotes the verse in Daniel 2: 11 which describes the angels. Though they may be present among men there "dwelling is not with those of flesh." Similar is the presence of the soul in man.