The interaction of He with metastable CO(a (3)Pi) gives rise to two adiabatic potential surfaces of reflection symmetry A' and A" which were calculated with the partially spin-restricted open-shell single and double excitation coupled cluster method with perturbative triples, RCCSD(T). Two diabatic potentials were constructed and fitted analytically; the appropriate form of the angular expansion functions was derived from general invariance properties. From variational calculations on these diabatic potential surfaces we obtained the quasibound vibration-rotation-spin levels of the CO-He complex in its lowest triplet state. Only the lower spin-orbit levels of this complex with approximate quantum number Omega = 0 of the CO(a (3)Pi) monomer were found to be stable with respect to dissociation into He and triplet CO. The potential and the bound van der Waals levels of the ground state CO(X (1)Sigma(+)) - He complex were recalculated and used in combination with the triplet excited state wave functions to compute the line strengths and the bound-bound part of the singlet-triplet excitation spectrum of the CO-He complex. The spin-forbidden singlet-triplet transitions access mainly the higher spin-orbit levels with \Omega\ = 1, but these were found to undergo rapid predissociation. The companion Paper II explicitly studies this process, predicts the excited state lifetimes, and generates the bound-continuum part of the CO-He singlet-triplet spectrum. (C) 2003 American Institute of Physics.