We study Zeeman splitting of zone-center subband edges in a cylindrical hole wire subject to a magnetic field parallel to its axis. The g factor turns out to fluctuate strongly as a function of wire subband index, assuming values that differ substantially from those found in higher-dimensional systems. We analyze the spin properties of hole-wire states using invariants of the spin-3/2 density matrix and find a strong correlation between g-factor value and the profile of hole spin-polarization density. Our results suggest possibilities for confinement engineering of hole spin splittings.