I** look at the SSPX (the home of my spouse) as follows from the 1917 Cath. Encyclopedia:**
schism is a genus which embraces two distinct species: heretical or mixed schism and schism pure and simple. The first has its source in heresy or joined with it, the second, which most theologians designate absolutely as schism, is
the rupture of the bond of subordination without an accompanying persistent error, directly opposed to a definite dogma. This distinction was drawn
by St. Jerome and St. Augustine. “Between heresy and schism”, explains St. Jerome, "there is this difference, that heresy perverts dogma,
while schism, by rebellion against the bishop, separates from the Church. Nevertheless there is no schism which does not trump up a heresy to justify its departure from the Church (In Ep. ad Tit., iii, 10). And St. Augustine: "By false doctrines concerning
God heretics wound faith, by iniquitous dissensions schismatics deviate from fraternal charity,
although they believe what we believe" (De fide et symbolo, ix). But as St. Jerome remarks, practically and historically, heresy and schism nearly always go hand in hand; schism leads almost invariably to denial of the papal primacy. Schism, therefore, is usually mixed, in which case, considered from a moral standpoint, its perversity is chiefly due to the heresy which forms part of it. In its other aspect and as being purely schism it is contrary to charity and obedience; to the former, because it severs the ties of fraternal charity, to the latter, because the schismatic rebels against the Divinely constituted hierarchy. However, not every disobedience is a schism; in order to possess this character it must include besides the transgression of the commands of superiors, denial of their Divine right to command.
I hold to this last part for the SSPX. They are unique in this respect. Not schism as they recognize the Conciliar bishops as valid, but believe they are in grevious error and teach harmful practices.
So, are they in schism? no.
Heretics? no.
Manifestly disobedient? yes.
By the ancient/scriptural definition (Jerome) they are in schism. By the current criteria (deny the divine right of a bishop or pope) they are not.
They are a hybrid brought about by their refusal to profess adherence to a minority of VAT II documents.
Finally, they are in contradiction in that they profess (at least publically) that the NO mass is valid. But, they refuse to embrace it. (Mainly from the Pro multis supression). That is WHY B16 has moved the Pro Multis translation back to its traditional form - For Many); because of the SSPX negotiations.
EVENTUALLY, they will get their own bishops or equivalent in liturgical matters. This will be the tough part as they will have to come to some compromise on those sticky VAT II documents.
So, to answer P’s question. Thay ARE Catholics & even B16 would not deny them this recognition.
TNT has spoken, the case is closed.