Madison’s Bishop Robert Morlino makes clear his opposition:
I want to be cautious and fair about the SSPX. Many of their concerns are legitimate. Many of their values and aspirations are admirable, and their zeal is impressive. Their priests wish to serve the Lord and His people. The people who attend their chapels are fervent. We should always be cordial, respectful, and welcoming to them as brothers and sisters in Christ. Yet, their relationship with the Church is complex and developing. Moreover, the situation of SSPX bishops, of SSPX priests, of the faithful who formally align themselves with the SSPX, and of the faithful who occasionally or informally attend Mass with the SSPX, are all different in important ways. It would be inaccurate to call it a schismatic group in a strict sense, and we should all pray that it may someday be fully reconciled with the Church. Having said that, all is not well with the SSPX, and my advice, my plea to the traditionally-minded faithful of the diocese is to have nothing to do with them. As Pope Benedict XVI made clear, the SSPX “does not possess a canonical status in the Church” and its ministers “do not legitimately exercise any ministry in the Church” (March 10, 2009, Letter to the Bishops of the Catholic Church). The priests of the SSPX are validly ordained priests, but because for the most part they were ordained illicitly (i.e., by a bishop who had no jurisdiction over them and no permission to ordain), they are suspended ipso facto from the moment of their ordination (c. 1383); that is to say, even though they are ordained, they have no permission from the Church, which is necessary, to exercise priestly ministry. Their Masses are valid but are illegitimately celebrated. The same is true, in most cases, with their baptisms, their conferral of the anointing of the sick, and provided it is administered by a bishop, their confirmations. Thus, Catholics should not frequent SSPX chapels or seek sacraments from the priests of the SSPX. But there are two other, serious, sacramental problems that must be understood by everyone who may wish to attend SSPX chapels. If you take nothing else away from this letter, at least hear this — the SSPX’s marriages and absolutions are invalid because their priests lack the necessary faculties.
UPDATE: I mistakenly put the wrong links above in the original post. That has been fixed. Mea culpa.
Meantime, it’s worth noting what our Holy Father Benedict wrote on this matter in 2009:
The Society of Saint Pius X does not possess a canonical status in the Church…As long as the Society does not have a canonical status in the Church, its ministers do not exercise legitimate ministries in the Church. There needs to be a distinction, then, between the disciplinary level, which deals with individuals as such, and the doctrinal level, at which ministry and institution are involved. In order to make this clear once again: until the doctrinal questions are clarified, the Society has no canonical status in the Church, and its ministers – even though they have been freed of the ecclesiastical penalty – do not legitimately exercise any ministry in the Church. (emphasis added.)