Both. The SSPX acknowledge Pope Francis as their Pope. They pray for the Holy Father at every Mass. I spent a month at a large SSPX chapel in North Idaho. The priests taught the Catholic faith clearly, without ambiguity. But in the time there, I never once heard anything remotely close to bashing Pope Francis. Not from the priests. Not from the laity. A lot of concern of this or that encylical or off the cuff remark, but nothing declaring him an anti christ or heretic (which is what I was expecting prior to going there since, in my ignorance, I bought into the anti sspx rhetoric going around). But the first thing I noticed when walking through the front doors of the Church was a big painting of Pope Francis. They accept that the Second Vatican Council was a legitimate council. They contest some of the things the council allegedly teaches and reject much of the errors being spread “in the spirit of vatican 2” after the council concluded. So the SSPX are within the fold of the Roman Catholic Church, they always have been. And whether or not they are disobedient is a matter of perspective. John Paul II saw them as disobedient and excommunicated yet-to-be-recognized St Marcel Lefebvre and the Bishops he ordained. Yet Pope Emeritus Benedict XVI saw that this was a mistake and lifted the excommunications on all the Bishops except the late Marcel Lefebvre (who had already passed on to his eternal reward). And between Pope Benedict and Pope Francis, all the SSPX Sacraments (which were always valid) have now become licit (in the eyes of the Roman hierarchy, though the SSPX makes a solid argument citing canon law that their sacraments were always licit as well). But for those that simply refuse to acknowledge the SSPX, there’s no reason to be concerned anymore as even in the eyes of the Roman hierarchy they are completely licit.
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