  We demonstrate that two-dimensional two-component bright solitons of an
annular shape, carrying vorticities $(m,\pm m)$ in the components, may be
stable in media with the cubic-quintic nonlinearity, including the
\textit{hidden-vorticity} (HV) solitons of the type $(m,-m)$, whose net
vorticity is zero. Stability regions for the vortices of both $(m,\pm m)$ types
are identified for $m=1$, 2, and 3, by dint of the calculation of stability
eigenvalues, and in direct simulations. A novel feature found in the study of
the HV solitons is that their stability intervals never reach the (cutoff)
point at which the bright vortex carries over into a dark one, hence dark HV
solitons can never be stable, contrarily to the bright ones. In addition to the
well-known symmetry-breaking (\textit{external}) instability, which splits the
ring soliton into a set of fragments flying away in tangential directions, we
report two new scenarios of the development of weak instabilities specific to
the HV solitons. One features \textit{charge flipping}, with the two components
exchanging the angular momentum and periodically reversing the sign of their
spins. The composite soliton does not split in this case, therefore we identify
such instability as an \textit{intrinsic} one. Eventually, the soliton splits,
as weak radiation loss drives it across the border of the ordinary strong
(external) instability. Another scenario proceeds through separation of the
vortex cores in the two components, each individual core moving toward the
outer edge of the annular soliton. After expulsion of the cores, there remains
a zero-vorticity breather with persistent internal vibrations.
