  In the rapidly evolving field of structured light, the self-torque has been
recently defined as an intrinsic property of light beams carrying
time-dependent orbital angular momentum. In particular, extreme-ultraviolet
(EUV) beams with self-torque -- exhibiting a topological charge that
continuously varies on the subfemtosecond timescale -- are naturally produced
in high-order harmonic generation (HHG) when driven by two time-delayed intense
infrared vortex beams with different topological charges. Until now, the
polarization state of such EUV beams carrying self-torque has been restricted
to linear states due to the drastic reduction in the harmonic up-conversion
efficiency with increasing the ellipticity of the driving field. In this work,
we theoretically demonstrate how to control the polarization state of EUV beams
carrying self-torque, from linear to circular. The extremely high sensitivity
of HHG to the properties of the driving beam allows us to propose two different
driving schemes to circumvent the current limitations to manipulate the
polarization state of EUV beams with self-torque. Our advanced numerical
simulations are complemented with the derivation of selection rules of angular
momentum conservation, which enable precise tunability over the angular
momentum properties of the harmonics with self-torque. The resulting high-order
harmonic emission, carrying time-dependent orbital angular momentum with a
custom polarization state, can expand the applications of ultrafast
light-matter interactions, particularly in areas where dichroic or chiral
properties are crucial, such as magnetic materials or chiral molecules.
