The spin up-down splitting of collective flows in intermediate-energy heavy-ion collisions as a result of the nuclear spin-orbit interaction is investigated within a spin-and isospin-dependent Boltzmann-Uehing-Uhlenbeck transport model SIBUU12. Using a Skyrme-type spin-orbit coupling quadratic in momentum, we found that the spin splittings of the directed flow and elliptic flow are largest in peripheral Au+Au collisions at beam energies of about 100-200 MeV/nucleon, and the effect is considerable even in smaller systems especially for nucleons with high transverse momenta. The collective flows of light clusters of different spin states are also investigated using an improved dynamical coalescence model with spin. Our study can be important in understanding the properties of in-medium nuclear spin-orbit interactions once the spin-dependent observables proposed in this work can be measured.