OK, let’s be technical.
Religious “orders” in the commonly used sense of the term (i.e. as “college” is used for “university” or “institute”).
Which is a part of the problem; people call Societies of Apostolic Life “orders” but they aren’t. And that’s actually a big deal, because the actual religious orders have a little thing called the “right of exemption”. Which, I may add, was being trampled on by Bishops for decades and centuries leading up to the Second Vatican Council. The right of exemption means that the Bishop can’t touch them, and isn’t in charge of them. Oh sure, the Bishop can remove them from diocesan ministries, but he cannot kick them out or discipline them.
Yet they were doing just that. We had Dominicans and Franciscans strongarmed to live outside their communities, and breaking their vows of poverty, for example. We had orders become clerical when they really should have been. Example; Franciscans don’t ordain everyone. They actually ordain very few. But they were being forced to ordain several people, and wouldn’t you know when Blessed John Paul II enforced the documents that V2 produced on this matter the Bishops listened.
Our local Archbishop sends our seminarians to another city, largely because their formation in our city wouldn’t be as good due to the teachers at the school. Could he send them t the local Dominican run school? He’d have to ask first. And even then, they’d get a DOMINICAN education, learn about community living. This is not a good thing for a secular priest who’ll be living on his own.
The SSPX is not an order, they are secular. At the end of the day, they
must respect and submit t the local Bishop. So too does the FSSP the Companions of the Cross, and others.
The Dominicans, all branches of them, do not report to the Bishop. Including the Lay Dominicans. When I’m done my formation and I get approval to start a ministry in a diocese, only my superior can kick me and my wife out of the place.
Ironically, the documents concerning religious orders were VERY traditional (and Traditional), yet a few traditionalists seem to dislike it.