  With the help of high-resolution long-slit and integral-field spectroscopy
observations, the number of confirmed cases of galaxies with counterrotation is
increasing rapidly. The evolution of such counterrotating galaxies remains far
from being well understood. In this paper we study the dynamics of
counterrotating collisionless stellar disks by means of $N$-body simulations.
We show that, in the presence of counterrotation, an otherwise gravitationally
stable disk can naturally generate bending waves accompanied by strong disk
heating across the disk plane, that is in the vertical direction. Such
conclusion is found to hold even for dynamically warm systems with typical
values of the initial vertical-to-radial velocity dispersion ratio $\sigma_{\rm
z}/\sigma_{\rm R} \approx 0.5$, for which the role of pressure anisotropy
should be unimportant. We note that, during evolution, the $\sigma_{\rm
z}/\sigma_{\rm R}$ ratio tends to rise up to values close to unity in the case
of locally Jeans-stable disks, whereas in disks that are initially
Jeans-unstable it may reach even higher values, especially in the innermost
regions. This unusual behavior of the $\sigma_{\rm z}/\sigma_{\rm R}$ ratio in
galaxies with counterrotation appears not to have been noticed earlier. Our
investigations of systems made of two counterrotating components with different
mass-ratios suggest that even apparently normal disk galaxies (i.e., with a
minor counterrotating component so as to escape detection in current
observations) might be subject to significant disk heating especially in the
vertical direction.
