  We present stable bright solitons built of coupled unstaggered and staggered
components in a symmetric system of two discrete nonlinear Schr\"{o}dinger
(DNLS) equations with the attractive self-phase-modulation (SPM) nonlinearity,
coupled by the repulsive cross-phase-modulation (XPM) interaction. These mixed
modes are of a "symbiotic" type, as each component in isolation may only carry
ordinary unstaggered solitons. The results are obtained in an analytical form,
using the variational and Thomas-Fermi approximations (VA and TFA), and the
generalized Vakhitov-Kolokolov (VK) criterion for the evaluation of the
stability. The analytical predictions are verified against numerical results.
Almost all the symbiotic solitons are predicted by the VA quite accurately, and
are stable. Close to a boundary of the existence region of the solitons (which
may feature several connected branches), there are broad solitons which are not
well approximated by the VA, and are unstable.
