Structural characteristics, vibrational frequencies, and energies of ten isomers of acetylene compounds with the centered aluminum cluster Al13 and its analogues Al12M doped with 3d transition metal atoms (M = Ti-Ni) in the states of different multiplicity have been calculated by the density functional theory method. In addition to “coordinated” intermediates in which the C2H2 molecule is coordinated through its C-C bond to the M vertex, an M-Al edge, or a trigonal face of the [MAl2] cluster, “fragment” isomers have been considered in which the acetylene molecule is broken into fragments (C2H + H, CH + CH, H + CH + C, and 2P + 2H) differently inserted into the aluminum cage and enlarging it to Al12MC2. For most compounds, low-lying isomers have structures 1–4 (the C2H2 molecule is coordinated to an Al2M face), 1–5 (two CH fragments are added to adjacent Al2M faces), and 1–8 (with a five-coordinate C* atom). Structure 1–1, in which the C2H2 molecule is coordinated through the C-C bond to the M dopant is unstable against transformation into 1–4 with a low barrier. An isomer with unusual structure 1–9 has been localized in which two five-coordinate C* atoms built into the aluminum cage are located in adjacent quasi-planar tetragonal [MAl3] faces and are bonded to the central aluminum atom (Alc) through the fifth bonds. The substitution of electronegative substituents X= F and Cl for H atoms in isomers 1–8 and 1–9 makes the latter more basic and clearly more favorable. The five-coordinate C* atoms in them are able to add acceptor ligands of the BH3 and AlH3 type and to increase the coordination number of the carbon atom to six with a considerable decrease in energy. The trends in the change in structural characteristics and relative energies of isomers with a change in M dopants along the 3d series, electronegativity of X substituents, and electronic state multiplicity have been analyzed.