  It is well known that the sixth Painlev\'e equation $\PVI$ admits a group of
B\"acklund transformations which is isomorphic to the affine Weyl group of type
$\mathrm{D}_4^{(1)}$. Although various aspects of this unexpectedly large
symmetry have been discussed by many authors, there still remains a basic
problem yet to be considered, that is, the problem of characterizing the
B\"acklund transformations in terms of Riemann-Hilbert correspondence. In this
direction, we show that the B\"acklund transformations are just the pull-back
of very simple transformations on the moduli of monodromy representations by
the Riemann-Hilbert correspondence. This result gives a natural and clear
picture of the B\"acklund transformations. Key words: B\"acklund
transformation, the sixth Painlev\'e equation, Riemann-Hilbert correspondence,
isomonodromic deformation, affine Weyl group of type $\mathrm{D}_4^{(1)}$.
