I wanted to offer a clarification about the SSPX’s canonical status: the priestly society has always claimed that it never lost its regularity, but this is being denied (implicitly) by the Vatican.
This all began in 1974 when the liberal French Episcopacy started to actively persecute the SSPX due to the traditionally-formed priests that would be returning to their dioceses - who they would be forced to accept.
Through maneuvering with the French Secretary of State, Cardinal Villot, Bishop Maime of Lausanne, Geneva and Fribourg diocese sent a letter to Archbishop Lefebvre declaring that he was suppressing the SSPX. However, per Canon Law, such an act was now reserved to the Holy See, since the SSPX had been officially approved by the Congregation of the Clergy - furthermore, the Vatican had granted the Society to incardinate clergy and religious from other dioceses and congregations (the Gethsemane, KY Trappist, Fr. Urban Synder, was one of the first to take advantage of this permission). Thus the local bishop no longer had the authority to suppress the SSPX.
Meanwhile, a declaration that Archbishop Lefebvre was suspended a divinis was issued - again under illegitimate grounds.
Archbishop Lefebvre attempted to have his case heard by the Apostolica Signatura (the highest ecclesiastical tribunal), but Cardinal Villot told its prefect, Cardinal Stafka, that if he accepted the case, he would be sacked. The appeal (which Archbishop Lefebvre would have clearly won) was subsequently denied - a great injustice.
Since then, the Vatican has not spoken clearly about the SSPX’s canonical situation, though this is clear de jure. Hence why the SSPX speaks of a political regularization, that is, a return (or recognition) of what is owed in justice: that the SSPX is a religious congregation of the Roman Catholic Church in good standing.
The article,
“The Hot Summer of 1976” has some fascinating historical details related to these events.