  In this paper we describe the effect on quantum groups -- namely, both QUEA's
and QFSHA's -- of deformations by twist and by 2-cocycles, showing how such
deformations affect the semiclassical limit. As a second, more important task,
we discuss how these deformation procedures can be "stretched" to a new extent,
via a formal variation of the original recipes, using "quasi-twists" and
"quasi-2-cocycles". These recipes seemingly should make no sense at all, yet we
prove that they actually work, thus providing well-defined, more general
deformation procedures. Later on, we explain the underlying reason that
motivates such a result in light of the "Quantum Duality Principle", through
which every "quasi-twist/2-cocycle" for a given quantum group can be seen as a
standard twist/2-cocycle for another quantum group, associated to the original
one via the appropriate Drinfeld functor. As a third task, we consider standard
constructions involving $R$-(co)matrices in the general theory of Hopf
algebras. First we adapt them to quantum groups, then we show that they extend
to the case of "quasi-$R$-(co)matrices", and finally we discuss how these
constructions interact with the Quantum Duality Principle. As a byproduct, this
yields new special symmetries (isomorphisms) for the underlying pair of dual
Poisson (formal) groups that one gets by specialization.
