I know nothing of Orthodoxy besides that it used to be one with Catholicism. How can I know I am in the right half of the split?
This is the very reason for the primacy–to guarantee the unity of the Church and that it can be easily recognized. One See, that of Rome, is fixed like a rock, always within the true Church. Without it, it would be impossible to discern the true Church after a pure schism (communion is broken, but a common faith is maintained). Both would have equal claims.
The way I look at it, the EO Churches simply are not one as we profess in the Creed. Many of their theologians even admit that the an ecclesiology that acknowledges one universal Church necessitates the primacy, which is why they try and defend a purely Eucharistic ecclesiology (which ignores or even denies a universal Church, making each particular Church instead the full manifestation of the Church). They are simply a collection of particular Churches (a particular Church being defined as a bishop and flock celebrating a common Eucharist) that are separated from that one, universal Church.
They constantly get into situations where EO particular church A is in communion with B, B is in communion with C, but A and C are not in communion with each other (A=B=C≠A) (e.g. the current schism between Constantinople, Moscow, and some Ukrainian Churches and others; ROCOR’s situation generally; the Moscow-Contantinople schism in 1996; the Bulgarian schism of the 19th century when most patriarchates, but not Moscow, broke communion with the Bulgarian Churches; etc., etc.). How can one universal/catholic church simultaneously have some particular churches in communion with other particular churches, while other churches are separated from each other? That’s not unity. This can only make sense if there is a plurality of Churches–the “one” of the Creed is lacking–and without this oneness, the very concept of one catholic/universal Church becomes untenable.
This was illustrated perfectly by the recent pan-Orthodox Synod in Crete a few years back (or whatever it ultimately was classified as). It barely even got off the ground because EO Churches were threatening to boycott (and many did) because they were fighting with other Churches over who had jurisdiction over what. Despite the EO polemics about all bishops being equal, if you look at how that synod was explicitly organized and carried out, the bishops who participated in that synod did not do so as equal bishops of one Church, but as representatives of multiple, distinct, national Churches and patriarchates. What was sought was not a consensus of the bishops of one Church (or even a consensus of particular Churches), but rather a consensus of independent national Churches–which didn’t happen anyway.
If the EO Churches may also be said to be holy (valid sacraments, etc.) and apostolic (in the broad sense), they lack oneness and catholicity (which go hand in hand). Only the Catholic Church has all four marks in the Creed.