It's moments like these that really get me going as far as gas goes.
I love seeing others really get into the in-depth parts of gas beyond the pretty lights and cool plasma
Well done with the shady alignment

Probably better than what I could do by hand, I've sort of been spoiled with the mounts, holders, and mirrors available.
If I may, I will try to explain what's happening here in a "nutshell" format.
SO!
To begin, a standard HeNe has involves a gas tube, vacuum sealed, two electrodes, and two mirrors. When a power is applied to these electrodes, and a discharge forms between the tube, you get excited atoms as a result.
Initially, you have excited He atoms, as well as Ne atoms. Both of which have high energy electrons. To get this light that you see, the He atoms collide with the Ne, increasing the energy. After this, the electrons fall back to ground, falling through many other levels, and releasing photons in the process.
However, at any one point, there's not enough light being produced to be visible. You use these mirrors to thus trap the light, allowing it to build with every passing second, until the cavity is saturated with enough light as possible. This "amount" is determined by cavity length, bore width, mirror reflectivity, etc, etc. Though that's not really worth getting in to.
However, that's a TON of light being trapped inside the tube. Also, keep in mind that one mirror is ~99.9%+ reflective (high reflector) and the other is about 98.5% (the output coupler). This is a roughly standard figure set for the 632.8nm line.
However! With this tube, only one side has a mirror, generally the HR. The other side is a window that is nearly 100% transparent to light one one polarization (resulting in a polarized output). When one places an external HR with the same radius of center (ROC) then you can get a stable resonator with dozens of watts intracavity power.
The ROC is very important, and difficult to explain in words. Similarly to how we use lenses to focus our lasers to a point, the mirrors do the same thing. Though the mirror is concaved, so the light hits it, and reflects back and at one point, will focus.
In order for a resonator to be stable, the diameter of the beam inside these two mirrors must never exceed the diameter of the mirror itself. I.E. no light is lost. Thus the light will continuously expand, then focus, and expand, and focus. Think of it in slow motion with a rod of light shining at one mirror, reflecting, focusing to a point, unfocusing, then getting to the same diameter again, hitting the other mirror, and doing the same. Like a tennis ball. In this case, the ball gets smaller, and then smallest while over the net, then largest when it hits the racket, no larger than the racket, and then back again.
That was a pair of awful run-on sentences, but I think I made my point
Basically. Good mirrors = lots of light. Bad mirrors = little to no light.
All about math!
Also, PM me if you need my skype name. I can definitely explain this two you verbally if you want.