  We investigate the role of electric toroidal dipoles in the prototypical
ferroaxial materials NiTiO$_3$ and K$_2$Zr(PO$_4$)$_2$, which undergo
ferroaxial structural phase transitions of order-disorder and displacive type,
respectively. Using first-principles electronic structure theory, we compute
the evolution across the ferroaxial transitions of the local electric toroidal
dipole moments, defined both in terms of the vortices formed by local dipoles,
as well as as the cross product of orbital and spin angular momenta. Our
calculations confirm that the electric toroidal dipole acts as the order
parameter for these ferroaxial transitions and highlight the importance of
spin-orbit coupling in generating a non-zero atomic-site electric toroidal
dipole moment. We find that, while the ferroaxial phases of NiTiO$_3$ and
K$_2$Zr(PO$_4$)$_2$ preserve global inversion symmetry, they contain
inversion-symmetry-broken sub-units that generate vortices of local electric
dipole moments. In addition to causing the net electric toroidal dipole moment,
these vortices induce a hidden spin polarization in the band structure.
