  Bialgebroids (resp. Hopf algebroids) are bialgebras (Hopf algebras) over
noncommutative rings. Drinfeld twist techniques are particularly useful in the
(deformation) quantization of Lie algebras as well as underlying module
algebras (=quantum spaces). Smash product construction combines these two into
the new algebra which, in fact, does not depend on the twist. However, we can
turn it into bialgebroid in the twist dependent way. Alternatively, one can use
Drinfeld twist techniques in a category of bialgebroids. We show that both
techniques indicated in the title: twisting of a bialgebroid or constructing a
bialgebroid from the twisted bialgebra give rise to the same result in the case
of normalized cocycle twist. This can be useful for better description of a
quantum deformed phase space. We argue that within this bialgebroid framework
one can justify the use of deformed coordinates (i.e. spacetime
noncommutativity) which are frequently postulated in order to explain quantum
gravity effects.
