  We study the Zeeman spin-splitting in hole quantum wires oriented along the
$[011]$ and $[01\bar{1}]$ crystallographic axes of a high mobility undoped
(100)-oriented AlGaAs/GaAs heterostructure. Our data shows that the
spin-splitting can be switched `on' (finite $g^{*}$) or `off' (zero $g^{*}$) by
rotating the field from a parallel to a perpendicular orientation with respect
to the wire, and the properties of the wire are identical for the two
orientations with respect to the crystallographic axes. We also find that the
$g$-factor in the parallel orientation decreases as the wire is narrowed. This
is in contrast to electron quantum wires, where the $g$-factor is enhanced by
exchange effects as the wire is narrowed. This is evidence for a $k$-dependent
Zeeman splitting that arises from the spin-3/2 nature of holes.
