  To a depth two extension A | B, we associate the dual bialgebroids S := \End
{}_BA_B and T := (A \o_B A)^B over the centralizer R=C_A(B). In the set-up
where R is a subalgebra of B, which is quite common, two nondegenerate pairings
of S and T will define an anti-automorphism \tau of the algebra S. Making use
of a two-sided depth two structure, we prove that \tau is an antipode and S is
a Hopf algebroid of a type we call skew Hopf algebra.
  A final section discusses how \tau and the nondegenerate pairings generalize
to modules via the pi-method for depth two, and a certain derived mapping of
cochain complexes is nullhomotopic.
