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The answer to my question “There were either twelve sexes or none.” explains that there were actually either five sexes or none.

I remember being confused at the time how DNA (or similar genetic material) would work with multiple sexes, but where did the twelve come from?

Is it possible that in the intervening years I mixed it up with twelve-stranded DNA-like genetic material from the Cheela on Dragon's Egg?

There is no mention of twelve-stranded DNA in the book.

Cheela biology is described in some technical detail:

The cheela biophysicists would not determine the genetic coding mechanism for the cheela for dozens of generations, but when they did, both they and the humans would be surprised at how different it was. Because of the high temperatures on the neutron star that attempted to disrupt everything into random chaos, and the all-pervasive magnetic field that lined everything up along the magnetic field lines, the cheela genetic structure was a triply-redundant linear strand of complex nuclear molecules. As the duplicating enzymes would copy the genetic molecule, the check at each triply redundant site provided an automatically correcting copying mechanism; if one of the three linear strands had mutated, the copying enzyme would be governed by majority rule, and the new triple strand would have the mutation corrected. If two mutations had occurred and all three sites were different, the enzyme would self-destruct, taking the faulty gene with it. It was only when the two mutations were the same that an error was able to creep
through.

I don't know enough biology to understand all this, but it seems to be three strands rather than twelve.

On the other hand, the number twelve does show up a lot in more obvious parts of the cheela body. From the Technical Appendix:

The cheela use a base 12 number system (they have twelve eyes) and their next unit of time after the turn is a great of turns or 144 turns. They occasionally use a dozen turns, but it has never had the same significance as the week does to humans. A great of turns is 28.7 seconds, while a human year is 31.6 million seconds. The ratio of a human year to a cheela great of turns is 1.1 million to one.

From studying the history of the cheela we have learned that a cheela spends about 12 greats (six minutes) as a hatchling; 12 greats as a young apprentice, 30 greats (15 minutes) as a worker, 12 greats as an Old One tending eggs and hatchlings, then the rest of its life (maximum of 24 greats or 12 minutes) as an Aged One. All of these indications lead to the conclusion that the effective relative time scale between the cheela and humans is approximately one million to one.