This is my interpretation of the Jade Helm logo, now that we know all of its elements.

Let's first go over the elements...

Double arrows

The arrows represent 'attack from the air' or 'attack from above'.

Dagger

The dagger represents thrust, death or attack.

Wooden Clog

A Wooden Clog can also be known as a 'Sabot' - meaning to 'Sabotage' or to 'disrupt' or 'throw a spanner in the works'. In addition, a 'Sabot' in military terms is an object used to fire a projectile. Importantly, the Clog is opaque (not quite transparent, i.e difficult to see or intended to represent as something hidden).

My interpretation?

An object in space may hit the earth (the Double arrows) causing death and destruction (the Dagger), disrupting society as we know it (the Wooden Clog). This object is hidden or difficult to see (at least now) hence the opaque nature of the Clog.