  Most close-in planetary satellites are in synchronous rotation, which is
usually the stable end-point of tidal despinning. Saturn's moon Hyperion is a
notable exception by having a chaotic rotation. Hyperion's dynamical state is a
consequence of its high eccentricity and its highly prolate shape (Wisdom et
al. 1984). As many binary asteroids also have elongated secondaries, chaotic
rotation is expected for moons in eccentric binaries (\'Cuk and Nesvorn\'y,
2010), and a minority of asteroidal secondaries may be in that state (Pravec et
al. 2016). The question of secondary rotation is also important for the action
of the BYORP effect, which can quickly evolve orbits of synchronous (but not
non-synchronous) secondaries (\'Cuk and Burns, 2005). Here we report results of
a large set of short numerical simulations which indicate that, apart from
synchronous and classic chaotic rotation, close-in irregularly-shaped
asteroidal secondaries can occupy an additional, intermediate rotational state.
In this ``barrel instability'' the secondary slowly rolls along its long axis,
while the longest axis is staying largely aligned with the primary-secondary
line. This behavior may be more difficult to detect through lightcurves than a
fully chaotic rotation, but would likewise shut down BYORP. We show that the
binary's eccentricity, separation measured in secondary's radii and the
secondary's shape are all important for determining whether the system settles
in synchronous rotation, chaotic tumbling, or barrel instability. We compare
our results for synthetic asteroids with known binary pairs to determine which
of these behaviors may be present in the Near-Earth Asteroid binary population.
