First, "sliding the signal back and forth" adds a phase shift to every frequency of phi = 2 * pi * f * t, where phi is the phase shift in radians, f is the frequency in Hz, and t is the time in seconds. This shows that you only get a phase shift of "pi" at ONE FREQUENCY, so sliding the waveform back and forth can not provide inversion of anything but one single frequency.
Second, once again, a phase shift of a signal in complex space is exactly described by a multiplication of the signal by cos phi + i * sin phi, where phi is the phase shift. Please note that at only 0 and pi phase shift is this not a complex signal. The i is the 'imaginary number' representing the quadraphase signal.
This is why your head hurts so very much trying to do a 90 degree phase shift of a real signal. This is also why 'DC' makes your head ache, because the DC and Pi components of the FFT are STRICTLY REAL. After you multiply the DC and Pi components by anything but 1 or -1, they have an imaginary component. If you take the IFFT with complex numbers at zero and pi, you get a complex OUTPUT. This is related to the problem in Laplace transforms, by the way, where you have to have matching number of poles and zeros, but that's only for old analog people to grok.
Remember, the DC and Pi result of any FFT for a real input ARE BOTH REAL. ALWAYS. Every other element can be real or not, and the negative (second half) of the FFT will be the complex conjugate of the first half, for each frequency. (remember that "frequency goes UP until you get to pi, then back down.) This is not a "good idea" its the basic math.