  We consider generic rotating axially symmetric "dirty" (surrounded by matter)
black holes. Near-horizon circular equatorial orbits are examined in two
different cases of near-extremal (small surface gravity $\kappa $) and exactly
extremal black holes. This has a number of qualitative distinctions. In the
first case, it is shown that such orbits can lie as close to the horizon as one
wishes on suitably chosen slices of space-time when $\kappa \rightarrow 0$.
This generalizes observation of T.\ Jacobson Class. Quantum Grav. 28 187001
(2011) made for the Kerr metric. If a black hole is extremal ($\kappa =0$),
circular on-horizon orbits are impossible for massive particles but, in
general, are possible in its vicinity. The corresponding black hole parameters
determine also the rate with which a fine-tuned particle on the noncircular
near-horizon orbit asymptotically approaches the horizon. Properties of orbits
under discussion are also related to the Ba% \~nados-Silk-West effect of high
energy collisions near black holes. Impossibility of the on-horizon orbits in
question is manifestation of kinematic censorship that forbids infinite
energies in collisions.
