What did St. Pius X mean with his motto "to restore all things in Christ" and how does it relate to the SSPX's mission? And did you know the Society has a secondary title?
It is not unusual for a newly-elect pontiff to write his encyclical program with his first letter. Pius X in 1903 made no exception and in his encyclical, E Supremi Apostolatus, he let everyone know of his intentions, plain and sublime, apostolic and transparent:
Some will certainly be found who, measuring Divine things by human standards will seek to discover secret aims of Ours, distorting them to an earthly scope and to partisan designs. To eliminate all vain delusions for such, We say to them with emphasis that We do not wish to be, and with the Divine assistance never shall be aught before human society but the Minister of God, of whose authority We are the depositary. The interests of God shall be Our interest, and for these We are resolved to spend all Our strength and Our very life. Hence, should anyone ask Us for a symbol as the expression of Our will, We will give this and no other: 'To renew all things in Christ.'"
Just a few months later, the Holy Father wrote on the occasion of the 50th anniversary of the dogma of the Immaculate Conception another letter Ad Diem Illum:
But the first and chief reason, Venerable Brethren, why the fiftieth anniversary of the proclamation of the dogma of the Immaculate Conception should excite a singular fervor in the souls of Christians lies for us in that restoration of all things in Christ which we have already set forth in Our first Encyclical letter… Upon His Mother most holy as upon a foundation, the noblest after Christ, rises the edifice of the faith of all centuries."
St. Pius X is conspicuously connecting the two letters and the reign of Christ with that of His holy Mother, and he certainly had much in common with the spirit of St. Louis de Montfort in this, explicitly or implicitly.
It will come to no surprise that, when Archbishop Lefebvre produced statutes to his new Society of St. Pius X, he gave it as a second name, and that is the Society of the Apostles of Jesus and Mary. This is what he wrote:
The Society is placed especially under the patronage of Jesus, the High Priest. For Our Lord’s whole existence was and remains priestly and the Sacrifice of the Cross was the reason for His Incarnation."
Further he explained thus the use of Mary’s patronage. The Society:
is also under the aegis of Mary, Mother of the Priest par excellence, and through Him Mother of all priests, in whom she forms her Son."
There is little doubt that the spirit of the Church and of the true Churchmen is to prolong the great work of the Incarnation in time and space. The best way to achieve it is to use the same means which have been used the first time: the Blessed Mother is the great means to beget Christ in souls and “recapitulate” all things in Him.