Well, my memory is very clear on the point that each nut had all 6
numbers since my solution depended on that fact. But sure, if we can
answer your extended problem all the better.

You mention rotations as a symmetry of a solution, which reminds me of
another kind of symmetry that is relevant to exactly what is meant by
"a set of nuts". Clearly the values of the integers inscribed on the
pieces have no effect of the character of the solution. If someone
broke into your house one night an erased all the 1's and replaced
them with 2's and replaced all the 2's with 1's, they wouldn't have
damadged your puzzle in any way (ignoring that you may have memorized
the solution by number, but that's your fault for chosing a bad way to
remember the solution). So applying any of the 6! permutations of 6
things to the numbers leaves your "set of nuts" fixed (relative to
eachother). The manufacturers's choice of numbering is thus at least
partially arbitrary.