We present ab initio calculations of cross sections for projectile and target excitation occurring in the course of He⁺ + He collisions using a three-active-electron semiclassical nonperturbative approach. Intermediate impact energies ranging from 1 keV to 225 keV/u are considered. The results of our calculations agree well with available measurements for both projectile and target excitation in the respective overlapping energy regions. A comparison of our results with those of other theoretical calculations further demonstrates the importance of a nonperturbative approach that includes a sufficient number of channels. Furthermore, it is found that the cross sections for target excitation into singlet states show a valley centered at about 25 keV/u, resulting from competition with electron transfer to singlet projectile states. By contrast, the cross sections for target excitation into triplet states do not exhibit any such structures.