As shown earlier, a linear transformation with the same form for the spatial coordinates as the Lorentz transformation (LT), and that allows for time dilation, but leaves simultaneity invariant instead of the one-way speed of light, predicts the same results as the LT for the usual tests of special relativity. Such a transformation is allowed by general covariance. A complementarity between the invariance of the one-way speed of light and the invariance of simultaneity is discussed. Using this transformation, interpreted as involving external synchronization, it is shown that two frames moving uniformly with equal and opposite velocities, v and –v relative to a third inertial frame, in which clocks are synchronized so that the one-way speed of light is c, can be related by a Galilean-like transformation with a relative velocity 2v/(1-(v/c)²).These transformations do not form a group, hence the term “pseudo-Galilean” is used to distinguish them from the Galilean transformations. An analogy with the Sagnac effect is discussed, and consistency with the LT for stellar aberration, and the Doppler effect is shown. Implication of the above complementarity for the possible unification of quantum theory and gravitation is briefly discussed, as well as the inferred physical significance of general covariance.