The Vatican has issued a
FAQ which addresses many of the points speculated about here.
In short, the bishops are no longer excommunicated, but the bishops and priests are all suspended, so their liturgies and sacraments are still illicit.
They are “separated from the Church” but “not in schism,” whatever that means.
Now, here’s the puzzler:
normally, if a priest has his faculties suspended, he cannot offer public liturgies or sacraments. If he does, he’ll be excommunicated.
In this case, they’re being un-excommunicated (?), but they are still suspended. They are still operating their ministries.
Is this some kind of weird test of their humility? Basically, to fully comply with what the letter, and the FAQ, say, the SSPX would have to shut down all their operations and act as ordinary laity for a while.
On their website, they have some responses, and they welcome the Vatican’s invitation to dialogue.
What they say in the different documents at the above link is that Popes Paul VI, John Paul II and Benedict XVI have
all said there are huge problems in the Church, and that common sense shows there are huge problems in the Church. They acknowledge that they have theological problems with the Council, which they believe to be at the root of the problems in the Church. Of course, Pope Benedict holds that the Council has been misused and misinterpreted to promote all sorts of bad agendas.
The real question will be whether the SSPX accept that the Council itself has been misinterpreted.
And exactly what their positions
are remain confusing. In some palces on their website, they assert their loyalty to the Pope.
Then, in some sections of the website, they list their criticisms of Vatican II, the teachings of JPII, and even some of the teachings of B16.
Then they have this whole section that’s basically “Articles on positions that we don’t agree with at all.” Yet, unlike most websites, they don’t offer
refutations of these viewpoints. It’s like, “We don’t agree with the opinions on this page, but we want you to see what they’rea ll about, just so you can decide for yourself.” That’s where they have all the sedevacantist stuff, real outright Vatican II stuff, etc.
But the main objections of the SSPX have to do with
a) Liturgy: B16 agrees with them, mostly
b) Ecumenism: B16 agrees with them, mostly
c) _Lumen Gentium), particularly the infamous “responsible parenting” passage (which, reportedly, Cardinal Ottaviani vociferously opposed on the Council floor): not sure what B16 thinks, but a lot of “loyal to the Pope” laity and priests agree with them.
Most of the arguments raised by the SSPX were also raised by Cardinal Ottaviani, who was prefect of the Holy Office/CDF during the Council itself.