Semitoric integrable systems, or semitoric manifiolds, have been completely classified in terms of five symplectic invariants, but there are properties of these systems which, while encoded in the invariants, are not easy to extract. In this talk I introduce a new invariant of semitoric systems which we call a semitoric helix and I outline its constrction. Using this invariant we are able to see clearly the effect of a semitoric blowdown/blowup on the system. We are then able to completely classify minimal semitoric manifolds; those that do not admit a semitoric blowdown. This work is joint with Alvaro Pelayo and Daniel M. Kane.