  In four space-time dimensions, there exists a special infinite-parameter
family of chiral modified gravity theories. All these theories describe just
two propagating polarizations of the graviton. General Relativity with an
arbitrary cosmological constant is the only parity-invariant member of this
family. We review how these modified gravity theories arise within the
framework of pure-connection formulation. We introduce a new convenient
parametrisation of this family of theories by using certain set of auxiliary
fields. Modifications of General Relativity can be arranged so as to become
important in regions with large Weyl curvature, while the behaviour is
indistinguishable from GR where Weyl curvature is small. We show how the Kasner
singularity of General Relativity is resolved in a particular class of modified
gravity theories of this type, leading to solutions in which the fundamental
connection field is regular all through the space-time. There arises a new
asymptotically De Sitter region `behind' the would-be singularity, the complete
solution thus being of a bounce type.
