Aluminum monofluoride (AlF) is a suitable molecule for laser cooling and trapping. Such experiments require extensive spectroscopic characterization of the electronic structure. Two of the theoretically predicted higher-lying triplet states of AlF, the counterparts of the well-characterized D1Δ and E1Π states, had not been experimentally identified yet. We here report on the characterization of the d3Π (v = 0-6) and e3Δ (v = 0-2) states, confirming the predicted energetic ordering of these states (J. Chem. Phys. 1988, 88, 5715-5725), as well as of the f3Σ+ (v = 0-2) state. The transition intensity of the d3Π, v = 3 - a3Π, v = 3 band is negligibly small. This band gets its weak, unexpected rotational structure via intensity borrowing from the nearby e3Δ, v = 2 - a3Π, v = 3 band, made possible via spin-orbit and spin-rotation interaction between the d3Π and e3Δ states. This interaction affects the equilibrium rotational constants in both states; their deperturbed values yield equilibrium internuclear distances that are consistent with the observations. We determined the ionization potential of AlF to be 78,492(1) cm-1 by ionization from the d3Π state.