  Using exact relations between velocity structure functions (Hill, Hill and
Boratav, and Yakhot) and neglecting pressure contributions in a first
approximation, we obtain a closed system and derive simple order-dependent
rescaling relationships between longitudinal and transverse structure
functions. By means of numerical data with turbulent Reynolds numbers ranging
from $\Re_\lambda=320$ to $\Re_\lambda=730$, we establish a clear
correspondence between their respective scaling range, while confirming that
their scaling exponents do differ. This difference does not seem to depend on
Reynolds number. Making use of the Mellin transform, we further map
longitudinal to (rescaled) transverse probability density functions.
