Lots of photos, lots of useful information. Caltech even have their own snowflake creation machine, that, instead of electrostatically levitating the snowflakes as they grow, or using a vertical blower, applies an electric field to grow narrow ice-spikes, and then lets the snowflakes form at the spikes' tips (which means that the central mount is probaby rigidly aligned to the resulting flake with atomic precision, and doesn't seem to affect the growing process).
If you're in the UK, and you've mocked train companies for blaming their electrical locomotive failures on "the wrong kind of snow", well, it turns out that snow crystallisation has a slightly crazy dependency on both temperature and airborne water content, forming a range of very different shapes, from the classic branched hexagon "christmas card" forms, to hexagonal plates or long hexagonal tubes (snowflake chart).
The CalTech site explains the wide variety of snowflake forms by this temperature-dependence: the idea being that snowflakes form symmetrically because the conditions across the flake are the same at any given time, and that the extreme variety of shapes is a function of the varying environmental conditions that the whole snowflake experiences as it falls through different regions of sky. It might go through a "spiky dendrite" phase, then change temperature and start trying to grow plates, and then go back to "dendrite" mode, and the exact amount of time spent in these different phases then dictates the shape that emerges.
If the identical patterning of the arms is purely a result of the identical (varying) growing conditions across the whole flake, then we don't require any additional mechanism for regulating symmetry. In that case, we'll expect individual snowflakes to accumulate diverging asymmetries as they grow, due to gradients of temperature or water availability or light or airflow across the flake. This'd seem to make the formation of extremely regular crystals a bit unlikely.
But the CalTech site argues that actually, most natural snowflakes are pretty irregular, and that people generally overestimate the degree of symmetry because the artsy folks who photograph them (presumably including CalTech!) give a misleading impression by carefully selecting out the "best" (most regular) flakes to photograph and publish.
That explanation seems to be a bit at odds with the current suggestion of how triangular snowflakes form, though: if triangular snowflakes grow because of airflow over the flake creating an asymmetrical growing environment, breaking the hex pattern, then if there wasn't an additional internal regulating symmetry-mechanism, there'd be no obvious reason why the resulting aerodynamically-disfigured flake should have 120-degee rotational symmetry. Airflow and a moisture gradient flowing across the flake in one direction might allows a bilateral left-right symmetry for the two sides of the flake that are experiencing the same growing conditions ... it doesn't explain why the conditions at the leading point of the falling tri-flake (falling point-first) should be identical to that at the two trailing side-points, or why points on the sides of those two trailing spurs points should be equivalent, when the airflow is hitting them at different angles. If triangular flakes are due to sideways airflow, then it means that the flake seems to be fighting to retain some sort of symmetry despite significant asymmetrical disruptive forces that ought to be destroying it. That'd increase the odds of there being a significant internal symmetry mechanism in play.
Of course, it may be that our explanation of triangular snowflakes is simply wrong, that airflow isn't disrupting the hex pattern, and that instead chemical contamination (or some other factor) is causing the alternative triangular crystal structure. But that'd still mean that something in our current understanding of snowflakes is wrong or incomplete. Even if yesterday's wacky suggestion about the quantum mirage effect is midguided, we'd still not know why snowflake formation is so sensitive to environmental conditions, or what the (non-aerodynamic) explanation of triangular snowflakes might be.
So again, more research needed.
The Caltech site's debunking of "mysterious" causes of snowflake symmetry is in the "Myths and Nonsense section" at http://www.its.caltech.edu/~atomic/snowcrystals/myths/myths.htm . The page says that there aren't any special forces at work here regulating symmetry, that most snowflakes are asymmetrical and "rather ugly", and that the published examples (including the ones on the site) are atypical, because "not many people are interested in looking at the irregular ones". In other words, if you look through the published work, you get a misleading impression due to publication bias. Well, yes ... quite possibly. But since the idea of what counts as "significant" symmetry might be a bit subjective,and since the datasets aren't available for us to look at, it's difficult to take this as a definitive answer until there's been actual experimental testing done.
Water is wierd stuff, and it keeps catching us out. I remember when people used to debunk ice spikes as an obvious example of psudoscience, and now those are understood, studied, and have their own page on the CalTech site. A lot of "crazy" ideas about water do turn out to be just as dumb as they first appear, but a few turn out to be correct. The trouble is, it's not always immediately obvious which are which.