If we look at the pressure generated in free field from a flat piston in a baffle, we can calculate analytically calculate the pressure generated via the so-called First Rayleigh integral [Fourier Acoustics, E.G. Williams].
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The pivotal point here is the sign on the righthand side. With w(Q) being the outwards displacement in a point Q on the piston, it is clearly seen that for a piston radiating into free space (mass-like impedance), this displacement is in anti-phase with p(P); the pressure in a measurement point P. This is really all we need to see. There, in general, is a 180 degree phase difference between the outwards displacement of a loudspeaker and the resulting pressure. Which of course means that when it moves outwards, defined as a positive displacement, we clearly get a negative pressure!