We present an exact analytical solution of the fundamental system of quasi-one-dimensional spin-1 bosons with infinite delta repulsion. The eigenfunctions are constructed from the wave functions of noninteracting spinless fermions, based on Girardeau's Fermi-Bose mapping. We show that the spinor bosons behave like a compound of noninteracting spinless fermions and noninteracting distinguishable spins. This duality is especially reflected in the spin densities and the energy spectrum. We find that the momentum distribution of the eigenstates depends on the symmetry of the spin function. Furthermore, we discuss the splitting of the ground state multiplet in the regime of large but finite repulsion.