Does anyone know the mechanism behind this double helix cloud formation? 
This looks like m=2 swirling instability mode of the axisymmetric jet, but how could an axisymmetric jet form up in the sky like that?
 A: Helix clouds are quite uncommon - a Google search reveals only a few, oft repeated, images. There are several short discussions about them on various forums, the longest of which glances over a few interesting ideas including an attempt to quantify the evolution of a well-documented Muscovian example with time. 
It isn't a jet stream core - they are too small (and found in areas where the jet stream isn't). You need something capable of generating a long thin cloud, and something to spin it. The first part is easy: a contrail being the prime example (as well as several other common cloud types). The second part requires some phenomenon to generate a twisting cylinder of air, and there is one! Strong localised vertical shear generates horizontal vortices that tend to twist over their length. 
That covers a single helix cloud - but what about the double helices? Well, contrails from larger airplanes form a fine example of two distinct lines of cloud, which, under the scenario described in the previous paragraph would wrap around each other to form such a shape. 
Example over Mauna Kea, generated from ambient cirrostratus.

(source: ethantweedie.com) 
Example over Moscow, almost certainly generated from a fleeting contrail.

