We present a comprehensive study of interactions (collisions) between two-dimensional composite vector solitons carrying topological charges in isotropic saturable nonlinear media. We numerically study interactions between such composite solitons for different regimes of collision angle and report numerous effects which are caused solely by the "spin" (topological charge) carried by the second excited mode. The most intriguing phenomenon we find is the delayed-action interaction between interacting composite solitons carrying opposite spins. In this case, two colliding solitons undergo a fusion process and form a metastable bound state that decays after long propagation distances into two or three new solitons. Another noticeable effect is spin-orbit coupling in which angular momentum is being transferred from "spin" to orbital angular momentum. This phenomenon occurs at angles below the critical angle, including the case when the initial soliton trajectories are in parallel to one another and lie in the same plane. Finally, we report on shape transformation of vortex component into a rotating dipole-mode solitons that occurs at large collision angles, i.e., at angles for which scalar solitons of all types simply go through one another unaffected.