PiusV is correct, Bayard, as is NRx. Interestingly, however, if +Gregory were to arrive at my Greek Orthodox parish Sunday and approach for communion, the priest would give him, as he would, say, a Copt, the holy gifts. He cannot give them to a Latin, though I understand that a Latin priest would give them to me.
It has to do with the nature of what seemed to be Non Chalcedonian theology vis a vis that held by the rest of the Church at the time of the Council of Chalcedon. Orthodox and Oriental Orthodox theologians have now come to understand that the differences are in the way we talk about the natures of Christ rather than in what he hold to be dogmatically true. Thus, a Copt, for example, or an Armenian, can receive in an Orthodox Church if there are no Oriental Orthodox churches reasonably close. The ecclesiology of the Latin Church, probably more than any theological issues, is what keeps our bishops out of communion with the Latin bishops and thus prevents us Orthodox from communing Latins or receiving communion at a Latin Mass. On the other hand, I am quite confident that lightning would not strike if piusv received at an Orthodox Liturgy or I at a Latin Mass. I do think it would be inappropriate, however.
When the subject is communio in sacris in the Orthodox Church your mileage is likely to vary a bit. A lot depends on who is approaching for communion and in which jurisdiction it’s happening. In some parts of the Middle East EO’s and OO’s have been taking communion in each others’ parishes for centuries and nobody bats an eye. Here in the US there is a more or less informal tolerance among the big three jurisdictions, those being the Greek Orthodox Archdiocese, the Orthodox Church in America (OCA), and the Antiochians.
But it’s not universal. I am 99% sure that the Russians and the Serbians don’t go for it. And I am also pretty sure that the Jerusalem Patriarchate doesn’t, although their presence here in the States is pretty thin. The JP has had notoriously chilly relations with the OO’s over disputes about access to the various shrines in the Holy land. And of course there is no concelebration among the clergy.
As for Catholics, the doctrinal differences are much more profound and Catholics are not normally communed by us. I have a hard time in this country imaging a situation where a Catholic might be in danger of death and the only priest available was Orthodox. But I suppose that given a true case of in articulo mortis we would hear confession and give communion. At least I would hope so. File that under the heading of Oikonomia.
The only exception that I am aware of, is among Maronite Catholics in Lebanon and that tragic country we used to call Syria. For cultural reasons that seem to trump doctrine, the local Orthodox and Maronite Catholics have been quietly intercommuning since forever. But again there is no concelbration among the clergy.