A bi-Galois object A is a bicomodule algebra for Hopf-Galois coactions with trivial invariants. In the spirit of Milnor's construction, we define the join of noncommutative bi-Galois objects (quantum torsors). To ensure that the diagonal coaction on the join algebra of the right-coacting Hopf algebra is an algebra homomorphism, we braid the tensor product A\ot A with the help of the left-coacting Hopf algebra. Our main result is that the diagonal coaction is principal. Then we show that an anti-Drinfeld double is a symmetric bi-Galois object with the Drinfeld-double Hopf algebra coacting on both left and right. In this setting, we consider a finite quantum covering as an example. Finally, we take the noncommutative torus with the natural free action of the classical torus as an example of a symmetric bi-Galois object equipped with a *-structure. It yields a noncommutative deformation of a nontrivial torus bundle.

Ludwik Dąbrowski was partially supported by the PRIN 2010-11 grant "Operator Algebras, Noncommutative Geometry and Applications" and WCMCS (Warsaw). He also gratefully acknowledges the hospitality of ESI (Vienna), IHES (Bures-sur-Yvette) and IMPAN (Warsaw). Tom Hadfield was financed via the EU Transfer-of-Knowledge contract MKTD-CT-2004-509794. Piotr M. Hajac was partially supported by NCN grant 2011/01/B/ST1/06474. Elmar Wagner was partially sponsored by WCMCS, IMPAN (Warsaw) and CIC-UMSNH (Morelia).