  We develop a general classification of the infinite number of families of
solitons and soliton complexes in the one-dimensional
Gross-Pitaevskii/nonlinear Schrodinger equation with a nonlinear lattice
pseudopotential, i.e., periodically modulated coefficient in front of the cubic
term, which takes both positive and negative local values. This model finds
direct implementations in atomic Bose-Einstein condensates and nonlinear
optics. The most essential finding is the existence of two branches of dipole
solitons (DSs), which feature an antisymmetric shape, essentially squeezed into
a single cell of the nonlinear lattice. This soliton species was not previously
considered in nonlinear lattices. We demonstrate that one branch of the DS
family (namely, the one which obeys the Vakhitov-Kolokolov criterion) is
stable, while unstable DSs spontaneously transform into stable fundamental
solitons (FSs). The results are obtained in numerical and approximate
analytical forms, the latter based on the variational approximation. Some
stable bound states of FSs are found too.
