  The complex structure of the valence band in many semiconductors leads to
multifaceted and unusual properties for spin-3/2 hole systems compared to
typical spin-1/2 electron systems. In particular, two-dimensional hole systems
show a highly anisotropic Zeeman spin splitting. We have investigated this
anisotropy in GaAs/AlAs quantum well structures both experimentally and
theoretically. By performing time-resolved Kerr rotation measurements, we found
a non-diagonal tensor $g$ that manifests itself in unusual precessional motion
as well as distinct dependencies of hole spin dynamics on the direction of the
magnetic field $\vec{B}$. We quantify the individual components of the tensor
$g$ for [113]-, [111]- and [110]-grown samples. We complement the experiments
by a comprehensive theoretical study of Zeeman splitting in in-plane and
out-of-plane fields $\vec{B}$. To this end, we develop a detailed multiband
theory for the tensor $g$. Using perturbation theory, we derive transparent
analytical expressions for the components of the tensor $g$ that we complement
with accurate numerical calculations based on our theoretical framework. We
obtain very good agreement between experiment and theory. Our study
demonstrates that the tensor $g$ is neither symmetric nor antisymmetric.
Opposite off-diagonal components can differ in size by up to an order of
magnitude.
