  Hydrodynamic equations for a binary mixture of inelastic Maxwell models
described by the Boltzmann equation are derived. The Navier-Stokes transport
coefficients are {\em exactly} obtained by solving the Boltzmann equation from
the Chapman-Enskog method for states close to the (local) homogeneous cooling
state (HCS). The knowledge of the transport coefficients allows one to analyze
two different problems. First, we solve the linearized hydrodynamic equations
around the homogeneous (cooling) state and identify the conditions for
stability as functions of the wave vector, the dissipation, and the parameters
of the mixture. As happens for monocomponent systems, the analysis shows that
the HCS is unstable to long enough wavelength perturbation. As a second
problem, we explore the validity of Onsager's reciprocal relations of a
granular binary mixture. As expected, since a granular system is not time
reversal invariant, Onsager's reciprocal relations do not apply for inelastic
collisions. The results show that the absence of the Gibbs state
(non-Maxwellian behavior of the velocity distribution functions describing the
HCS), the collisional cooling, and the occurrence of different kinetic
temperatures for both species (breakdown of energy equipartition) are
responsible for a violation of Onsager's relations
