  We investigate the dynamics of $f(R)$ gravity in Jordan and Einstein frames.
First, we perform a phase-space singularities analysis in both frames. We show
that, typically, anisotropic singularities are absent in the Einstein frame,
whereas they may appear in the Jordan frame. We conciliate this apparent
inconsistency by showing that the necessary conditions for the existence of the
Einstein frame are namely the same ones assuring the absence of the anisotropic
singularities in the Jordan frame. In other words, we show that, at least in
the context of Bianchi I cosmologies, the Einstein frame is available only when
the original formulation in the Jordan frame is free of anisotropic
singularities. Furthermore, we present a novel dynamical system formulation for
anisotropic cosmologies in which both frames, provided they exist, will be
manifestly equivalent from the dynamical point of view, even though they fail
to be diffeomorphic in general. Our results could help not only the
construction of viable (free of anisotropic singularities) $f(R)$ cosmological
models, but also contribute to the still active debate on the physical
interpretation of the two frames.
