  We study zero-energy collisions between three identical bosons with spin $f =
1$ interacting via pairwise potentials. We quantify the corresponding
three-body scattering hypervolumes, which parametrize the effective three-body
interaction strengths in a many-body description of spin-1 Bose-Einstein
condensates. Our results demonstrate universal behavior of the scattering
hypervolumes for strong $s$- and $p$-wave two-body interactions. At weak
interactions we find that the real parts of the scattering hypervolumes are
predominantly determined by hard-hyperspherelike collisions which we
characterize by a simple formula. With this universal result we estimate that
spin mixing via three-body collisions starts to dominate over two-body spin
mixing at a typical particle density of $10^{17}~\mathrm{cm}^{-3}$ for
${}^{23}$Na and ${}^{41}$K spinor condensates. This density can be reduced by
tuning the two-body interactions to an $s$- or $p$-wave dimer resonance or to a
point where two-body spin mixing effectively vanishes. Another possibility to
observe effects of three-body spin mixing involves the application of weak
magnetic fields to cancel out the effective two-body interaction strength in
the characteristic timescale describing the spin dynamics.
