The dynamics of vertically compensated two-layer vortices (hetons) with finite cores are examined within the context of the quasi-geostrophic approximation on the f-plane. The two-layer version of the contour dynamics method is used, and complemented by the so-called contour surgery technique. Special attention is paid to two-heton interactions when the initial locations of the vortex patches are symmetrical. A classification of the different regimes observed is made according to external parameters, namely geometrical parameters and stratification. In this parameter space, novel quasi-stationary states resulting from collisions between two hetons may be observed: (i) formation of a configuration consisting of two-layer vortices moving in opposite directions and, as a special case, a configuration analogous to the ‘klapstoss’ billiard shot, (ii) absorption of one heton by the other and subsequent movement of a new dipole, (iii) formation of two-layer dipoles, larger than the original hetons, associated with rotating peripheral satellite eddies in their wakes. Some of these results may have implications for the analysis of mesoscale vortices in the ocean.