  We show that the Ehresmann-Schauenburg bialgebroid of a quantum principal
bundle $P$ or Hopf Galois extension with structure quantum group $H$ is in fact
a left Hopf algebroid $L(P,H)$. We show further that if $H$ is
coquasitriangular then $L(P,H)$ has an antipode map $S$ obeying certain minimal
axioms. Trivial quantum principal bundles or cleft Hopf Galois extensions with
base $B$ are known to be cocycle cross products $B\#_\sigma H$ for a
cocycle-action pair ($\vartriangleright$,$\sigma$) and we look at these of a
certain `associative type' where $ \vartriangleright$ is an actual action. In
this case also, we show that the associated left Hopf algebroid has an antipode
obeying our minimal axioms. We show that if $L$ is any left Hopf algebroid then
so is its cotwist $L^\varsigma$ as an extension of the previous bialgebroid
Drinfeld cotwist theory. We show that in the case of associative type,
$L(B\#_\sigma H,H)=L(B\# H)^{\tilde\sigma}$ for a Hopf algebroid cotwist
$\varsigma=\tilde\sigma$. Thus, switching on $\sigma$ of associative type
appears at the Hopf algebroid level as a Drinfeld cotwist. We view the affine
quantum group $\hat{U_q(sl_2)}$ and the quantum Weyl group of $u_q(sl_2)$ as
examples of associative type.
