  We use well-resolved direct numerical simulations of high-Reynolds-number
turbulence to study a fundamental statistical property of turbulence -- the
asymmetry of velocity increments -- with likely implications on important
dynamics. This property, ignored by existing small-scale phenomenological
models, manifests most prominently in the non-monotonic trend of velocity
increment moments (or structure functions) with the moment order, and in
differences between ordinary and absolute structure functions for a given
separation distance. We show that high-order structure functions arise nearly
entirely from the negative side of the probability density of velocity
increments, essentially removing the ambiguity between ordinary and absolute
moments, and provide a plausible dynamical interpretation of this result.
