Official relations between the two groups? In a word, nonexistent, owing to the exclusivity claims of each side. Understandable.
As I like to say, a rite is a package deal, and I think the core constituencies of both groups, born members of their faiths whose respective rites influence their whole lives (admirable), are so immersed in those rites that they really haven't got the time to look into other rites and churches. (In the cases of the Russian and Greek constituencies of the Orthodox churches and splinter groups, this holisticity - is there such a word? - is also bound up with ethnic identity.)
But ISTM that such Catholics who are aware of such things, like conservative Roman Catholics in general who are hip to the existence of the Christian East, are sympathetic to the Eastern Churches (as JoeZollars has said) and see them as an inspiration and working model of much of what they want. (The same faith but with a mystical bent all its own. Yes, I know that's not how the Orthodox see themselves!) This is possible owing to Catholic belief about these churches, including splinter groups outside Orthodoxy, being holy and apostolic - 'valid' bishops and Eucharist as well as a basic shared orthodoxy. Of course they see the other side as schismatic - there's the claim to be the one true church, almost mirroring Eastern Orthodoxy's teaching about itself.
But such also often (but not always) think, at the same time (!) that their rite is 'more Catholic' or 'universal'. Pr+Ãƒâ€šÃ‚Âªstantia ritus latini - Roman equals Catholic. (A stick with which Byzantine Catholics have been beaten for centuries.) Disowned as official policy but it affects policy and attitudes to this day.
My impression is most members of the Catholic splinter groups, like many RCs of all kinds, are only vaguely aware the Orthodox exist. (And probably don't know anything about the calendar war in the East.) As occasional poster Brendan03 puts it, Protestantism is their historic Other - other churches aren't on their radar.
Again, as for the Eastern side (including the ex-Orthodox splinter groups), I don't think their ethnic congregations know much at all about other churches and aren't particularly interested in them. Those members not born into it (namely converts) who do know about them usually are anything but sympathetic - more often they are obsessed the other way, vehemently anti as a number of ranting, embarrassing online writings attest.
One thing some 'trads' and the East share, on top of basic apostolic beliefs, is an emphasis on custom - organic development of rites and the authority of doing things as they always have been done. (Other trads, however, are different, using narrowly legalistic arguments based on their reading of canon law to defend their liturgical practices.)