In the same spirit and with the same commitment to encouraging the resolution of all fractures and divisions in the Church and to healing a wound in the ecclesial fabric that was more and more painfully felt, I wished to remit the excommunication of the four Bishops illicitly ordained by Archbishop Lefebvre. With this decision I intended to remove an impediment that might have jeopardized the opening of a door to dialogue and thereby to invite the Bishops and the "Society of St. Pius X" to rediscover the path to full communion with the Church. As I explained in my Letter to the Catholic Bishops of last 10 March, the remission of the excommunication was a measure taken in the context of ecclesiastical discipline to free the individuals from the burden of conscience constituted by the most serious of ecclesiastical penalties. However, the doctrinal questions obviously remain and until they are clarified the Society has no canonical status in the Church and its ministers cannot legitimately exercise any ministry. (emphasis added)
The SSPX bishops were excommunicated and their illicit consecrations defined as schismatic in the motu proprio Ecclesia Dei issued July 2, 1988. The excommunications have been lifted, but the canonical status of the SSPX remains unchanged.
There are two important points here:
1. The SSPX is not in full communion with the Church and is invited by the Church to rediscover this path.
2. The SSPX has no canonical status in the Church and its ministers cannot legitimately exercise any ministry.
This has never been formally changed by the Holy See, in spite of Pope Francis' recent indults allowing the Society to hear confessions — a benefit provided explicitly for the good of the faithful in the Jubilee Year of Mercy, and extended indefinitely.
Communion with the Chair of Peter is a constitutive part of Roman Catholicism — something the SSPX accepts in principle but rejects in practice. As Pope Benedict said, the SSPX has "no canonical status," "no legitimate ministry," is "invited to rediscover the path to full communion," has no faculties from any bishop, is not part of the divinely ordained hierarchical structure of the Church and all its priests are suspended a divinis.
We are well aware of ongoing dialogue between the SSPX and Rome. It is to be fervently hoped that this dialogue results in a return of the SSPX to full communion with the Church, granting its bishops and clergy canonical status and the authority to exercise ministry. Reconciliation of the SSPX with the Church would be a great blessing for the Church but most especially for the SSPX. The SSPX has consistently refused the most generous and magnanimous invitations from the Church to reconcile and place itself under the authority of the Vicar of Christ, rejecting his dogmatically defined "full power of shepherding, ruling and governing the universal Church," a power "ordinary and immediate over all the churches and over each and every member of the faithful" ("The Pope Exercises Supreme Jurisdiction" [Original Italian here]).
Perhaps the SSPX fears it will be crushed by an unsympathetic hierarchy and pope. Perhaps it is justified in such fearful apprehensions. But there is precedent in the history of the Church — in the lives of the saints, in the life of Our Lord Himself — for risking or allowing oneself to be crushed by obedience. Until such time as the SSPX chooses to reconcile with the Church, Church Militant must support the judgment of the Holy Father and the Church concerning its status.
"My Journey out of the Lefebvre Schism" (external website)