Equatorial circular orbits of test particles in the Kerr-anti-de Sitter black-hole and naked-singularity spacetimes are analyzed and their properties like the existence, orientation and stability are discussed. Due to the attractive cosmological constant (), all particles moving along equatorial orbits are still bound in the gravitational field of the central object. In general, there are two families of equatorial circular orbits. Particles moving along minus-family orbits possess negative angular momentum and, thus, they are counterrotating from the point of view of the locally non-rotating frames (LNRF). Particles moving along plus-family orbits possess, in most cases, positive angular momentum and belong to corotating particles from the point of view of the LNRF. Nevertheless, in stationary regions inside black holes and also near naked singularities with appropriately chosen value of the cosmological constant and rotational parameter , there are also counterrotating plus-family circular orbits. Moreover, in spacetimes with , some of these orbits are characterized by negative specific energy, indicating the bounding energy of the particle, moving along such an orbit, higher than its rest energy. In black-hole spacetimes, all such orbits are radially unstable, but in naked-singularity spacetimes, stable counterrotating orbits with negative specific energy exist.