  We derive transition rules for Kasner exponents in bouncing Bianchi I models
with generic perfect fluid matter fields for a broad class of modified gravity
theories where cosmological singularities are resolved and replaced by a
non-singular bounce. This is a generalization of results obtained previously in
limiting curvature mimetic gravity and loop quantum cosmology. A geometric
interpretation is provided for the transition rule as a linear map in the
Kasner plane. We show that the general evolution of anisotropies in a Bianchi I
universe -- including during the bounce phase -- is equivalent to the motion of
a point particle on a sphere, where the sphere is the one-point
compactification of the Kasner plane. In addition, we study the evolution of
anisotropies in a large family of bouncing Bianchi I space-times. We also
present a novel explicit solution to the Einstein equations for a Bianchi I
universe with ekpyrotic matter with a constant equation of state $\omega=3$.
