The asymmetry comes from the different masses of electrons and neon ions (neon ions are about 36000 times heavier).
This mass asymmetry results in different cross sections for the excitation of neon atoms by electrons and ions. There are some plots of this here http://webhost.ua.ac.be/plasma/pdf_papers/sab97comparison.pdf (figure 1a for electrons, figure 1b for ions). The interesting processes of excitation occur above around 10eV for electrons and 100eV for ions.
There is a very cool Java simulation of discharges: http://phet.colorado.edu/en/simulation/discharge-lamps . It covers electron excitation, the atomic structure of neon, acceleration and excitation cross-sections very well, and demonstrates why the glow can be localized (it glows at the point that electrons have been accelerated to the necessary energy), and why it is asymmetric (all electrons start from the cathode and can gain energy on the way to the anode).
I hope this answers why only one side can glow. I wish I could explain exactly why the glow is next to the cathode in the picture, but I would just be guessing.