  We present a Lax pair for the sixth Painlev\'e equation arising as a
continuous isomonodromic deformation of a system of linear difference equations
with an additional symmetry structure. We call this a symmetric
difference-differential Lax pair. We show how the discrete isomonodromic
deformations of the associated linear problem gives us a discrete version of
the fifth Painlev\'e equation. By considering degenerations we obtain symmetric
difference-differential Lax pairs for the fifth Painlev\'e equation and the
various degenerate versions of the third Painlev\'e equation.
