The linear molecular ions H₂He⁺, HHe⁺₂, and He⁺₃ are the central units (chromophores) of certain He-solvated complexes of the H₂He⁺_n, HHe⁺_n, and He⁺_n families, respectively. These are complexes which do exist, according to mass-spectrometry studies, up to very high n values. Apparently, for some of the H₂He⁺_n and He⁺_n complexes, the linear symmetric tetratomic H₂He⁺₂ and the diatomic He⁺₂ cations, respectively, may also be the central units. In this study, definitive structures, relative energies, zero-point vibrational energies, and (an)harmonic vibrational fundamentals, and, in some cases, overtones and combination bands, are established mostly for the triatomic chromophores. The study is also extended to the deuterated isotopologues D₂He⁺, DHe⁺₂, and D₂He⁺₂. To facilitate and improve the electronic-structure computations performed, new atom-centered, fixed-exponent, Gaussian-type basis sets called MAX, with X = T(3), Q(4), P(5), and H(6), are designed for the H and He atoms. The focal-point-analysis (FPA) technique is employed to determine definitive relative energies with tight uncertainties for reactions involving the molecular ions. The FPA results determined include the 0 K proton and deuteron affinities of the ⁴He atom, 14 875(9) cm⁻¹ [177.95(11) kJ mol⁻¹] and 15 229(8) cm⁻¹ [182.18(10) kJ mol⁻¹], respectively, the dissociation energies of the He⁺₂ → He⁺ + He, HHe⁺₂ → HHe⁺ + He, and He⁺₃ → He⁺₂ + He reactions, 19 099(13) cm⁻¹ [228.48(16) kJ mol⁻¹], 3948(7) cm⁻¹ [47.23(8) kJ mol⁻¹], and 1401(12) cm⁻¹ [16.76(14) kJ mol⁻¹], respectively, the dissociation energy of the DHe⁺₂ → DHe⁺ + He reaction, 4033(6) cm⁻¹ [48.25(7) kJ mol⁻¹], the isomerization energy between the two linear isomers of the [H, He, He]⁺ system, 3828(40) cm⁻¹ [45.79(48) kJ mol⁻¹], and the dissociation energies of the H₂He⁺ → H⁺₂ + He and the H₂He⁺₂ → H₂He⁺ + He reactions, 1789(4) cm⁻¹ [21.40(5) kJ mol⁻¹] and 435(6) cm⁻¹ [5.20(7) kJ mol⁻¹], respectively. The FPA estimates of the first dissociation energy of D₂He⁺ and D₂He⁺₂ are 1986(4) cm⁻¹ [23.76(5) kJ mol⁻¹] and 474(5) cm⁻¹ [5.67(6) kJ mol⁻¹], respectively. Determining the vibrational fundamentals of the triatomic chromophores with second-order vibrational perturbation theory (VPT2) and vibrational configuration interaction (VCI) techniques, both built around the Eckart–Watson Hamiltonian, proved unusually challenging. For the species studied, VPT2 has difficulties yielding dependable results, in some cases even for the fundamentals of the H-containing molecular cations, while carefully executed VCI computations yield considerably improved spectroscopic results. In a few cases unusually large anharmonic corrections to the fundamentals, on the order of 15% of the harmonic value, have been observed.