... that rule about the initial jerk always being in the direction of the move ...
That's just calculus: the second derivative is always in the same direction.
It's the same with distance and acceleration, if you want to move a positive distance you need to start with a positive acceleration.
So if you have a positive velocity you need an initial positive jerk.
It gets more complicated if you are already moving, e.g. if you are tracing a profile, where you might already have negative acceleration and negative velocity happening and the next segment is "more positive" then you will have a positive addition to the existing negative values of V and A , and J will be positive (having been zero prior to this) .