Why does a balloon spiral in the air instead of moving in a straight line? When an air-filled balloon is released without its opening tied up, it moves in a circular path rather than a straight line. Why is that? 
 A: If you make a cardboard tube, put this into the ballons nozzle and then let go you'll find the balloon goes in a mostly straight line. It probably won't go exactly straight because the balloon probably isn't exactly cylindrically symmetrical, but it will go a lot straighter than without the cardboard tube. I recall doing this in primary school long before I'd even heard of physics let alone learnt any.
Bearing this in mind, I would guess the usual erratic flight is because the ballon nozzle is flexible and the airflow through it is turbulent, so the nozzle orientation changes randomly as the air flows out through it.
A: A friend sent this as his explanation and it seems quite satisfactory to me:
For a balloon to fly in a straight line, the direction of the jet of expelled air would have to be in line with the balloon’s centre of mass and its centre of drag – the point where the forces resisting the balloon’s forward motion are symmetrical
If these two centres don’t coincide, the centre of drag should be behind the centre of mass, otherwise stability is compromised. The reason darts and arrows have flights is to keep the two centres in line and ensure drag is at the rear of the moving projectile.
If the balloon’s line of thrust does not pass through the centre of mass (which is almost certain) but is in the same plane as the line joining the centres of mass and drag (which
is unlikely), the balloon would travel in a circle in that plane, although the pull of gravity will ultimately force it down to the ground, especially as the air driving it forward expires. However, because these lines generally do not intersect, thrust from the balloon’s opening comes at an angle to the plane of the circle, pushing the balloon into the helical, screw-like motion you saw when carrying out the experiment. The thrust of the balloon and the air resistance to the balloon will not cancel each other out in such a situation and so a turning moment is exerted.
