Relationships are derived between pressure–velocity–velocity (PVV) statistics, fourth-order velocity structure functions, and the pressure structure function. The PVV statistics are related to the correlation of pressure at one point with the product of two velocity components at another point. The Navier–Stokes equation, isotropy, and incompressibility are used; no other assumption is used. Thus the relationships apply for all Reynolds numbers and can be used as a benchmark to determine how well turbulence models mimic pressure fluctuations in Navier–Stokes turbulence. A necessary condition limiting compressibility is given. The inertial-range and viscous-range formulas of the PVV statistics are obtained. The results are compared with previous theories that used the joint Gaussian approximation. Data from grid turbulence are used to evaluate the statistics.