Is it a valid general statement that avalanche rectifiers used in series for higher voltage do NOT need equalizing resistors? I haven't used them before in this way and I want to verify.
What makes avalanche rectifiers different is that they are fabricated to have a very uniform junction so the entire junction breaks down at the same time instead of just selected spots which could lead to localized overheating and damage. So they can be reliably used in series without balancing but as MagicSmoker points out, thermal constraints still need to be considered. There is probably some derating rule for operating them this way but offhand I do not know it.
The best rule is "use big ones"

Running rectifiers well below their max ratings looks like overkill, but it usually ensures long life.
A case in point is the really large ones NEC used for the HT supplies of their transmitters.
Over decades associated with that equipment, I have never encountered a rectifier failure, or even heard of one.
Pye, back when they still made transmitters (mid to late 1980s) used much smaller rectifiers, supposedly within their ratings.
They would "blow up" if you looked sideways at them.
We went through so many of them, & the supply time from the UK was so long & slow, we joked that it would almost have been easier to send someone to the UK to pick up replacements from the factory.