  We consider a two-dimensional (2D) two-component spinor system with cubic
attraction between the components and intra-species self-repulsion, which may
be realized in atomic Bose-Einstein condensates, as well as in a
quasi-equilibrium condensate of microcavity polaritons. Including a 2D
spatially periodic potential, which is necessary for the stabilization of the
system against the critical collapse, we use detailed numerical calculations
and an analytical variational approximation (VA) to predict the existence and
stability of several types of 2D symbiotic solitons in the spinor system.
Stability ranges are found for symmetric and asymmetric symbiotic fundamental
solitons and vortices, including hidden-vorticity (HV) modes, with opposite
vorticities in the two components. The VA produces exceptionally accurate
predictions for the fundamental solitons and vortices. The fundamental
solitons, both symmetric and asymmetric ones, are completely stable, in either
case when they exist as gap solitons or regular ones. The symmetric and
asymmetric vortices are stable if the inter-component attraction is stronger
than the intra-species repulsion, while the HV modes have their stability
region in the opposite case.
