A slight nit to pick here, John; there's "mixing" in the sense that you use it here, and then there's modulation based on multiplication, which may be something else entirely. A "mixer", meaning the specific sort of circuit that this term is generally applied to, does preserve the original frequencies. However, a "balanced mixer" does not, and the difference is that the latter actually DOES perform an operation that is solely a multiplication. In mathematical terms:
sin(A) * sin(B) = 1/2[cos(A-B) - cos(A+B)]
which clearly doesn't have the original frequencies in the output. So pure multiplication results in DSBSC, which is then often converted to SSB by filtering out one of the sidebands. To get the original carrier back, a constant term has to be introduced, i.e.,
sin(A) * [K + sin(B)]
Here, the K term corresponds to the DC offset which exists when , for instance, the modulating signal (here, sin(B)) is applied to the carrier (sin (A)) via a modulation transformer (classically, in the "plate" circuit of a class C output stage), as is the case very commonly in commercial AM installations. The result is conventional "AM", with a "full" carrier.
Just to clarify that for those who may be trying to follow the terminology, since this IS sci.electronics.basics after all. I know YOU know this stuff already...
Bob M.