We present a study of the complex phase-lag behavior of the low-frequency (<20 Hz) quasi-periodic oscillations (QPOs) in the X-ray transient and black hole candidate XTE J1550-564 during its very high state. We distinguish two different types of low-frequency QPOs, based on their coherence and harmonic content. The first type is characterized by a 6 Hz QPO with a Q (the QPO frequency divided by the QPO width) of less than 3 and with a harmonic at 12 Hz. The second type of QPO is characterized by a 6 Hz QPO with a Q-value of greater than 6 and with harmonics at 3, 12, 18, and possibly at 9 Hz. Not only are the Q-values and the harmonic content of the two types different, but their phase-lag behavior also differs. For the first type of QPO, the low-energy photons (<5 keV) of both the 6 Hz QPO and its harmonic at 12 Hz lag the hard energy photons (>5 keV) by as much as 1.3 rad. The phase lags of the second type of QPO are more complex. The soft photons (<5 keV) of the 3 and 12 Hz QPOs lag the hard photons (>5 keV) by as much as 1.0 rad. However, the soft photons of the 6 Hz QPO precede the hard ones by as much as 0.6 rad. This means that different harmonics of this type of QPO have different signs for their phase lags. This unusual behavior is hard to explain when the lags are due to light-travel time differences between the photons at different energies, e.g., in a Comptonizing region surrounding the area in which the QPOs are formed.