I can only talk about my personal experience with the Jesuits and clam no general knowledge about the order.
Two years ago, I earned my diploma in theology at a Jesuit-lead university in Germany and I am currently working on my doctorate in moral theology there. (We have an English webpage, too!
sankt-georgen.de/default/english.html )
It also houses a seminary for several German dioceses (Limburg an Hamburg, namely).
In my experience, it is true when Jesuits are called “liberal”, and I know many people who frequent this forum don’t like it. But I’d like to make a distinction between a “good” kind of liberals in the church and “bad” ones. “Good” ones: Not particularily attached to traditions with a small “t” and open to try out new things, but faithful to the magisterium. “Bad” ones: Trying new things for the sake of novelty and not caring about being faithful.
I experienced the Jesuits as the good kind of liberals. Not cassock-wearing, most not even collar-wearing, not celebrating the most traditional masses, but never irreverent and open for discussion about everything and very educated indeed - but not questioning the authority of the magisterium, at least not in my hearing, and not diverting from it.
I have to concede two exceptions, I heard one professor, who is also a memeber of the SJ, say that he thinks the arguments against a female priethood weak and aner one who thought it wrong to forbid artificial conception altogether, but both of them added that this was their private opinion only.
I think this is not much for spending four years with them, and unfortunately, I have heard much worse from Non-SJ-students.