We introduce left and right groups of bisections of a Hopf algebroid and show that they form a group crossed homomorphism with the group $Aut(\mathcal{ L})$ of bialgebroid automorphisms. We also introduce a nonAbelian cohomology $H^2(\mathcal{ L},B)$ governing cotwisting of a Hopf algebroid with base $B$. We also introduce a notion of coquasi-bialgebroid $\mathcal{ L}$ via a 3-cocycle on $\mathcal{ L}$. We also give dual versions of these constructions. For the Ehresmann-Schauenburg Hopf algebroid $\mathcal{ L}(P,H)$ of a quantum principal bundle or Hopf-Galois extension, we show that the group of bisections reduces to the group $Aut_H(P)$ of bundle automorphisms, and give a description of the nonAbelian cohomology in concrete terms in two cases: $P$ subject to a `braided' commutativity condition and $P$ a cleft extension or `trivial' bundle. Next we show that the action bialgebroid $B\# H^{ op}$ associated to a braided-commutative algebra $B$ in the category of $H$-crossed (or Drinfeld-Yetter) modules over a Hopf algebra $H$ is an fact a Hopf algebroid. We show that the bisection groups are again isomorphic and can be described concretely as a natural space $Z^1_{ \triangleleft}(H,B)$ of multiplicative cocycles. We also do the same for the nonAbelian cohomology and for $Aut(\mathcal{ L})$. We give specific results for the Heisenberg double or Weyl Hopf algebroid $H^*\# H^{ op}$ of $H$. We show that if $H$ is coquasitriangular then its transmutation braided group $B=\underline{ H}$ provides a canonical action Hopf algebroid $\underline H\# H^{ op}$ and we show that if $H$ is factorisable then $\underline H\# H^{ op}$ is isomorphic to the Weyl Hopf algebroid of $H$. We also give constructions for coquasi versions of $\mathcal{ L}(P,H)$ and of the Connes-Moscovici bialgebroid. Examples of the latter are given from the data of a subgroup $G\subseteq X$ of a finite group and choice of transversal.
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