  We study coupled unstaggered-staggered soliton pairs emergent from a system
of two coupled discrete nonlinear Schr\"{o}dinger (DNLS) equations with the
self-attractive on-site self-phase-modulation nonlinearity, coupled by the
repulsive cross-phase-modulation interaction, on 1D and 2D lattice domains.
These mixed modes are of a "symbiotic" type, as each component in isolation may
only carry ordinary unstaggered solitons. While most work on DNLS systems
addressed symmetric on-site-centered fundamental solitons, these models give
rise to a variety of other excited states, which may also be stable. The
simplest among them are antisymmetric states in the form of discrete twisted
solitons, which have no counterparts in the continuum limit. In the extension
to 2D lattice domains, a natural counterpart of the twisted states are vortical
solitons. We first introduce a variational approximation (VA) for the solitons,
and then correct it numerically to construct exact stationary solutions, which
are then used as initial conditions for simulations to check if the stationary
states persist under time evolution. Two-component solutions obtained include
(i) 1D fundamental-twisted and twisted-twisted soliton pairs, (ii) 2D
fundamental-fundamental soliton pairs, and (iii) 2D vortical-vortical soliton
pairs. We also highlight a variety of other transient dynamical regimes, such
as breathers and amplitude death. The findings apply to modeling binary
Bose-Einstein condensates, loaded in a deep lattice potential, with identical
or different atomic masses of the two components, and arrays of bimodal optical
waveguides.
