Yes, I suppose a slight change in the docs is in order; terming it a bug is rather extreme, tho.
To assuage one of your objections, let us use a manifestly unsymmetric function:
f[w_] := Exp[i w] UnitStep[w]
Now, evaluate the following in sequence:
InverseFourierSequenceTransform[f[w], w, n]
FourierCoefficient[f[w], w, n, FourierParameters -> {1, -1}]
FourierCoefficient[f[w], w, n]
InverseFourierSequenceTransform[f[w], w, n, FourierParameters -> {1, -1}]
Stare at the four expressions just generated for a full minute.
Having done so, go back to the documentation for InverseFourierSequenceTransform[] and FourierCoefficient[]. Note well the bullet points under "Details and Options" the description for the FourierParameters option for both functions. Compare and contrast.
After all that, if you still feel motivated to complain to Support, you should now be able to write a more refined report armed with the information you gained from doing that suggested activity.
