Numerical simulations show that episodic or prolonged accretion of gas from environment (Thakar & Ryden 1996, 1998) and merging with a gas-rich dwarf companion (Thakar & Ryden 1996, Thakar et al. 1997) are viable mechanisms for the retrograde acquisition of small amounts of external gas. They give rise to counter-rotating gaseous disks only in S0 galaxies, since in spiral galaxies the acquired gas is swept away by the pre-existing gas. Therefore, the formation of counter-rotating gaseous disks is favored in S0 galaxies since they are gas-poor systems, while spiral disks host large amounts of gas (Bettoni et al. 2003), which is corotating with the stellar component. When gas-rich systems acquire external gas in retrograde orbits, the gas clouds of the new retrograde and pre-existing prograde components collide, lose their centrifugal support, and accrete toward the galaxy center. A counter-rotating gaseous disk will be observed only if the mass of the newly supplied gas exceeds that of the pre-existing one (Lovelace & Chou 1996). A counter-rotating stellar disk is the end-result of star formation in the counter-rotating gas component. For this reason we observe a larger fraction of counter-rotating gaseous disks in S0s than in spirals (Pizzella et al. 2004). This also explains why the mass of counter-rotating gas in most S0 galaxies is small compared to that of the stellar counter-rotating components (Kuijken et al. 1996). Counter-rotating gaseous and stellar disks in spirals are both the result of retrograde acquisition of large amounts of gas, and they are observed with the same frequency (Pizzella et al. 2004). In this framework, stellar counter-rotation is the end result of star formation in a counter-rotating gaseous disk. The formation of two counter-rotating stellar disks from material accreted from two distinct filamentary structures in cosmological simulations has been recently discussed by Algorry et al. (2014).
Usually, major mergers between disk galaxies with comparable masses are ruled out because they tend to produce ellipticals. Anyway, for a narrow range of initial conditions, major mergers are successful in building a remarkably axisymmetric disk which hosts two counter-rotating stellar components of similar mass and size (Puerari & Pfenniger 2001, see also Bettoni et al., this volume]. Moreover, the coplanar merging of two counter-rotating progenitors heats more the prograde than the retrograde stellar disk and the gas ends up aligned with the total angular momentum (dominated by the orbital angular momentum), and thus with the prograde stellar disk, as observed in NGC 4550 (Crocker et al. 2009).
An alternative to the external-origin scenarios has been proposed by Evans & Collett (1994) for NGC 4550, and it involves the dissolution of a bar or a triaxial stellar halo. In this process, the stars moving on box orbits escape from the confining azimuthal potential well and move onto tube orbits. In non-rotating disks, there are as many box orbits with clockwise azimuthal motion as with counter-clockwise. Thus, half box-orbit stars are scattered onto clockwise-streaming tube orbits, half onto counter-clockwise ones. In this way, two identical counter-rotating stellar disks can be built.
Since barred galaxies host quasi-circular retrograde orbits, the origin of the stellar counter-rotation observed in barred galaxies (Bettoni 1989, Bettoni & Galletta 1997) can be the result of internal dynamical processes (Wozniak & Pfenniger 1997). Nevertheless, accreted gas can be trapped on this family of retrograde orbits and then eventually form new stars. These counter-rotating material may lead to the formation of a secondary bar rotating in opposite direction with respect to the main one (Sellwood & Merritt 1994). Observationally, the formation of secondary bars may be constrained by addressing the occurrence of counter-rotating secondary bars. Indeed, the most accepted view on the origin of secondary bars is that they form through instabilities in gas inflowing along the main bar (Shlosman et al. 1989). But, a retrograde bar is unlikely to be supported by a prograde disk. Numerical simulations suggest that two counter-rotating nested bars, formed in two counter-rotating stellar disks that overlap each other, are stable and long-living systems (Friedli 1996). This leads to the possibility that secondary bars form out of inner stellar disks, like those observed in the nuclei of several disk galaxies (Pizzella et al. 2002, Ledo et al. 2010). To date counter-rotating nuclear disks have been detected only in elliptical galaxies (e.g., Morelli et al. 2004).