To a finite Hopf-Galois extension $A | B$ we associate
dual
bialgebroids $S := End\,_BA_B$ and $T := (A \o_B A)^B$ over the centralizer
$R$ using the depth two theory
in [18,Kadison-Szlachányi]. First we extend results on the equivalence of certain
properties of Hopf-Galois
extensions with corresponding properties of the coacting Hopf algebra [21,8]
to depth two extensions using coring theory [3]. Next we show
that $T^{\rm op}$ is a Hopf algebroid over the centralizer $R$
via Lu's theorem \cite[23,5.1] for smash products with special modules over the
Drinfel'd double, the Miyashita-Ulbrich action, the
fact that $R$ is a commutative algebra in
the pre-braided category of Yetter-Drinfel'd modules [28]
and the equivalence of Yetter-Drinfel'd modules with modules over
Drinfel'd double [24].
In our last section, an exposition of results of Sugano [29,30]
leads us to a Galois correspondence
between sub-Hopf algebroids
of $S$ over simple subalgebras of the centralizer with finite projective
intermediate simple subrings
of a finite projective H-separable extension of simple rings $A \supseteq B$.