  In [1] a new notion of Hopf algebroid has been introduced. It was shown to be
inequivalent to the structure introduced under the same name in [17]. We review
this new notion of Hopf algebroid. We prove that two Hopf algebroids are
isomorphic as bialgebroids if and only if their antipodes are related by a
`twist' i.e. are deformed by the analogue of a character. A precise relation to
weak Hopf algebras is given. After the review of the integral theory of Hopf
algebroids we show how a right bialgebroid can be made a Hopf algebroid in the
presence of a non-degenerate left integral. This can be interpreted as the
`half of the Larson-Sweedler theorem'. As an application we construct the Hopf
algebroid symmetry of an abstract depth 2 Frobenius extension [2].
