  Terahertz vortex beams with different superposition of the orbital angular
momentum $l={\pm}1$, ${\pm}2$, ${\pm}3$, and ${\pm}4$ and spin angular momentum
${\sigma}={\pm}1$ were used to study antiferromagnetic (AFM) resonances in
TbFe$_3$(BO$_3$)$_4$ and Ni$_3$TeO$_6$ single crystals. In both materials we
observed a strong vortex beam dichroism for the AFM resonances that are split
in external magnetic field. The magnitude of the vortex dichroism is comparable
to that for conventional circular dichroism due to ${\sigma}$. The selection
rules at the AFM resonances are governed by the total angular momentum of the
vortex beam: $j={\sigma}+l$. In particular, for $l={\pm}2$, ${\pm}3$, and
${\pm}4$ the sign of $l$ is shown to dominate over that for conventional
circular polarization ${\sigma}$.
