  In this paper, we revisit the claim that the Eulerian and quasi-Lagrangian
same time correlation tensors are equal. This statement allows us to transform
the results of an MSR quasi-Lagrangian statistical theory of hydrodynamic
turbulence back to the Eulerian representation. We define a hierarchy of
homogeneity symmetries between incremental homogeneity and global homogeneity.
It is shown that both the elimination of the sweeping interactions and the
derivation of the 4/5-law require a homogeneity assumption stronger than
incremental homogeneity but weaker than global homogeneity. The
quasi-Lagrangian transformation, on the other hand, requires an even stronger
homogeneity assumption which is many-time rather than one-time but still weaker
than many-time global homogeneity. We argue that it is possible to relax this
stronger assumption and still preserve the conclusions derived from theoretical
work based on the quasi-Lagrangian transformation.
