  The recently observed two and four-quasiparticle high-spin rotational bands
in the odd-odd nuclei $^{166, 168, 170, 172}$Re are investigated using the
cranked shell model with pairing correlations treated by a particle-number
conserving method. The experimental moments of inertia and alignments can be
reproduced well by the present calculation if appropriate bandhead spins and
configurations are assigned for these bands, which in turn confirms their spin
and configuration assignments. It is found that the bandhead spins of those two
rotational bands observed in $^{166}$Re~[Li {\it et al.}, Phys. Rev. C 92
014310 (2015)] should be both increased by $2\hbar$ to get in consistent with
the systematics of the experimental and calculated moments of inertia for the
same configurations in $^{168, 170, 172}$Re. The variations of the
backbendings/upbendings with increasing neutron number in these nuclei are
investigated. The level crossing mechanism is well understood by analysing the
variations of the occupation probabilities of the single-particle states close
to the Fermi surface and their contributions to the angular momentum alignment
with rotational frequency. In addition, the influence of the deformation
driving effects of the proton $1/2^-[541]$ ($h_{9/2}$) orbtial on the level
crossing in $^{172}$Re is also discussed.
