As I understand it, all torsion classes correspond to t-structures in the way described by Greg, but there is almost always more t-structures than torsion classes (even taking into account the shifts).

As I understand it, all torsion classes correspond to t-structures in the way described by Greg, but there is almost always more t-structures than torsion classes (even taking into account the shifts). I think t-structures are closer to some kind of filtration on the abelian category. There's a paper on stability conditions (Gorodentsev, A. L.; Kuleshov, S. A.; Rudakov, A. N. $t$-stabilities and $t$-structures on triangulated categories. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 4, 117--150; translation in Izv. Math. 68 (2004), no. 4, 749--781) that talks about this a bit.