I'll answer my own question:
The function f is real-valued iff the FT of f is Hermitian (conjugate symmetric).
f is Hermitian iff:
a) real part of f is even.
b) imaginary part of f is odd.
Summary: for a function f to be real:
a) The amplitude spectrum must have even symmetry.
b) The phase spectrum must have odd symmetry.
I'll answer my own question:
The function f is real-valued iff the FT of f is Hermitian (conjugate symmetric).
f is Hermitian iff:
a) real part of f is even.
b) imaginary part of f is odd.
Summary: for a function f to be real:
a) The amplitude spectrum must have even symmetry.
b) The phase spectrum must have odd symmetry.