  We propose that the bimodal nature of the kick velocity distribution of radio
pulsars is connected with the dichotomy between hadronic stars ({\it i.e.}
neutron stars with no quark matter content) and quark stars. Bimodality can
appear due to different mechanisms of explosion which leads to the formation of
two types of compact stars or due to two different sets of parameters mastering
a particular kick mechanism. The low velocity maximum (at $\sim 100$ km
s$^{-1}$) is connected with hadronic star formation, whereas the second peak
corresponds to quark stars. In the model of delayed collapse of hadronic stars
to quark stars (Berezhiani et al. 2003\nocite{bbd2003}) quark deconfinement
leads to a second energy release, and to a second kick, in addition to the kick
imparted to the newly formed hadronic star during the supernova explosion. If
the electromagnetic rocket mechanism can give a significant contribution to
pulsar kicks, then the high velocity peak can be connected with the shorter
initial spin periods of quark stars with respect to hadronic stars. We discuss
{\it pro et contra} of these scenarios.
