The alkali-metal trihalides MX3 (M = Li, Na, K, Rb, and Cs; X = Cl, Br, and I) are systematically studied using coupled-cluster methods. Benchmarks using CCSD(T) against diatomic experimental results suggest satisfactory performance for the weighted core-valence basis sets (new basis sets for K, Rb, and Cs) selected for predicting reliable structures and harmonic vibrational frequencies. An isomer search using the B3LYP functional yields a planar, yet asymmetric T-shaped C-s, structure as the global minimum for all MX3 species. Much higher level CCSD(T) computations show a moderate to strong distortion of the X-3(-) anion by the M+ cation in the respective equilibrium geometries. Most obviously, for LiCl3, the two Cl-Cl distances are separated by 0.786 angstrom. Even for CsI3, the structure least distorted from the M+X3- model, the two I-I distances differ by 0.243 angstrom. It does not take much energy to distort the parent anions along an antisymmetric stretch, so this is no surprise. The normal modes of vibration of the MX3 molecules are in better agreement with matrix isolation experiments than previous calculations. And these normal modes reveal that, instead of the well-established antisymmetric and symmetric stretches of the "free" X-3(-) anions, relatively localized and mutually perturbed X-X and M-X stretches are calculated. The suggestion emerges that the MX3 system may be alternatively described as an MX-X-2 complex rather than the MX3- ion pair. This perspective is supported by bonding analyses showing low electron densities at the bond critical points and natural bond orders between the MX and X-2 moieties. The thermochemistry of fragmentations of MX3 to MX + X-2 versus M+ + X-3(-) also supports the alternative viewpoint of the bonding in this class of molecules.