Fr. Steven Hawke-Steeples, S.J., a Ruthenian Jesuit liturgist
Throughout the world, but especially (I think) in the Americas, the Jesuits are heavily involved in the lives of Byzantine Catholics. This is not to say that other Roman religious orders do not play a role (after all, His Grace Bishop John Kudrick of Parma was originally a Third Order Franciscan), but I see the Jesuits having a predominantly large role among Roman Catholics assisting Byzantine Catholics.
What are some examples? From my own life, at the Ruthenian mission that I attend, there are five priests that assist the community: two of them are Jesuits. One of the great Byzantine liturgists of the present day is Archimandrite Robert Taft, S.J. A Polish-American priest being considered for sainthood for his mission work among Eastern Catholics in the Soviet Union is Servant of God Water Ciszek, S.J. The head of the Slovak Catholic Church is Metropolitan Ján Babjak, S.J. Then there's the fact that our Jesuit Holy Father, Pope Francis, was formerly the ordinary for all Eastern Catholics in Argentina. This is all not counting the many, many Jesuits who have worked with Byzantine Catholics throughout the centuries.
(As a side note, the Jesuit influence wasn't always perfectly positive: the Jesuits are somewhat infamous for pushing Latinizations on Eastern Catholics in the past. The Russian Orthodox also, in general, detest the Jesuits for the creation of the Ukrainian Catholic Church: according to a common Russian Orthodox view, this Eastern Catholic Church was formed by the conversion of Orthodox at musketpoint by the Jesuits.)
What makes the Jesuits so prevalent among Byzantine Catholics as compared to other religious orders? Is it the spirituality? The devotion to the name of Jesus is definitely a common thread, since the Byzantine Church loves the Jesus prayer so much. The discernment of spirits is important in both spiritualities. The dedication of "for the greater glory of God" (ad majorem Dei gloriam) in all things that the Jesuits have could be compared with the Byzantine focus on divinization, though I think there is a bit of contrast (doing things for the glory of God vs. becoming the glory of God, in a sense). The imaginative aspect of the Spiritual Exercises doesn't seem to mesh well with the disdain for the imagination in many Eastern spiritual writers. The obedience of the Jesuits, though, seems to connect pretty well with the Eastern love of their bishops.
Is it the spirituality that leads the Jesuits to the Byzantines? I think, rather, that it is that last aspect, their obedience, combined with their sense of mission. Jesuits make a special vow of obedience to the Holy Father: their goal is to fulfill the needs of the Church as the Holy Father wishes them to. Often, these means going to do mission work as, for instance, the great St. Francis Xavier did, or as Servant of God Matteo Ricci did. If the Holy Father sees mission work as a need of the Church, the Jesuits will often fulfill this role. That is how Servant of God Walter Ciszek ended up serving Byzantine Christians: there was a call from Rome for mission work in the Soviet Union, and Fr. Ciszek (because of his desire to always do what is hardest, as he would attribute it) answered the call.
I think it is this devotion to obedience and the love of mission that grew out of that that began the relation of the Jesuits and the Byzantines. But what about those Jesuits doing scholarship on Byzantine Christianity? Jesuits also have a love of scholarship, and maybe there was a sense of "mission" to fill the need of Byzantine Catholic scholarship. This is all just hypothetical on my part, but what is undeniable is this: as history as played out, the Jesuits are inseparable from the history and life of Byzantine Catholics, at least to some extent.
Servant of God Walter Ciszek, pray for us!