  We address vortex solitons supported by optical lattices featuring modulation
of both the linear and nonlinear refractive indices. We find that when the
modulation is out-of-phase the competition between both effects results in
remarkable shape transformations of the solitons which profoundly affect their
properties and stability. Nonlinear refractive index modulation is found to
impose restrictions on the maximal power of off-site solitons, which are shown
to be stable only below a maximum nonlinearity modulation depth.
