The properties of microswimmer dumbbells composed of pusher-puller pairs are investigated by mesoscale hydrodynamic simulations employing the multiparticle collision dynamics approach for the fluid. An individual microswimmer is represented by a squirmer, and various active-stress combinations in a dumbbell are considered. The squirmers are connected by a bond, which does not impose any geometrical restriction on the individual rotational motion. Our simulations reveal a strong influence of the squirmers' flow fields on the orientation of their propulsion directions, their fluctuations, and the swimming behavior of a dumbbell. The properties of pusher-puller pairs with an equal magnitude of the active stresses depend only weakly on the stress magnitude. This is similar to dumbbells of microswimmers without hydrodynamic interactions. However, for non-equal stress magnitudes, the active stress implies strong orientational correlations of the swimmers' propulsion directions with respect to each other, as well as the bond vector. The orientational coupling is most pronounced for pairs with large differences in the active-stress magnitude. The alignment of the squirmers' propulsion directions with respect to each other is preferentially orthogonal in dumbbells with a strong pusher and weak puller, and antiparallel in the opposite case when the puller dominates. These strong correlations affect the active motion of dumbbells, which is faster for strong pushers and slower for strong pullers.