We use the Bogoliubov-de Gennes formalism and study the ground-state phases of trapped spin-orbit-coupled Fermi gases in two dimensions. Our main finding is that the presence of a symmetric (Rashba-type) spin-orbit coupling spontaneously induces counterflowing mass currents in the vicinity of the trap edge, i.e., up arrow and down arrow particles circulate in opposite directions with equal speed. These currents flow even in noninteracting systems, but their strength decreases toward the molecular Bose-Einstein-condensate limit, which can be achieved by increasing either the spin-orbit coupling or the interaction strength. These currents are also quite robust against the effects of asymmetric spin-orbit couplings in the x and y directions, gradually reducing to zero as the spin-orbit coupling becomes one dimensional. We compare our results with those of chiral p-wave superfluids and superconductors.