One of my light bulbs recently failed, for the first time in a few years, and I had to go buy a replacement. Usually I try to look for the ones with the best EU energy label rating, since not only do more efficient bulbs use less power, but they also generate less waste heat and thus are likely to last longer (not to mention probably being of better construction in the first place).
I was used to typical energy labels for LED bulbs ranging from A to A++, but this time I got quite confused, since about half of the bulbs on the shelves had energy labels like E or F, which I'm more used to associating with the old incandescent "heat balls". Curious, I did some math and realized that some of those "E" rated bulbs actually had better lumens / watt ratios than the A+ rated ones!
What I started to suspect at that point, and eventually confirmed by Googling, is that the labeling system is being changed right now, with the new labels being quite a bit stricter than the old ones especially for things like light bulbs, which are nowadays a lot more efficient than 10 or 20 years ago. Confusingly, though, the new "rescaled" labels look a lot like the old ones and there's apparently an 18 month transition period, until the end of February 2023, during which products with either new or old labels can be sold in stores.
So far, so good. What I couldn't easily find an answer to, however, is how do the new labels compare with the old ones? That is, for example, if I see two LED light bulbs on the store shelf, one rated A+ under the old system and the other rated E under the new system, which one of those is actually more energy efficient? Or do I just have to do the math myself?
(I did eventually find what I think is the answer, but it involved digging into the EU standards and doing some non-trivial math, so I figured it was worth sharing. I'm posting my self-answer below, and I welcome any reviews, corrections and/or improvements to it, as well as any additional answers.)