  Legendre-Gauss-Lobatto (LGL) grids play a pivotal role in nodal spectral
methods for the numerical solution of partial differential equations. They not
only provide efficient high-order quadrature rules, but give also rise to norm
equivalences that could eventually lead to efficient preconditioning techniques
in high-order methods. Unfortunately, a serious obstruction to fully exploiting
the potential of such concepts is the fact that LGL grids of different degree
are not nested. This affects, on the one hand, the choice and analysis of
suitable auxiliary spaces, when applying the auxiliary space method as a
principal preconditioning paradigm, and, on the other hand, the efficient
solution of the auxiliary problems. As a central remedy, we consider certain
nested hierarchies of dyadic grids of locally comparable mesh size, that are in
a certain sense properly associated with the LGL grids. Their actual
suitability requires a subtle analysis of such grids which, in turn, relies on
a number of refined properties of LGL grids. The central objective of this
paper is to derive just these properties. This requires first revisiting
properties of close relatives to LGL grids which are subsequently used to
develop a refined analysis of LGL grids. These results allow us then to derive
the relevant properties of the associated dyadic grids.
