We have studied the interaction of carbon monoxide with a Re(10-10) surface between 150 and 800 K by means of low-energy electron diffraction (LEED), temperature-programmed thermal desorption, and work function change (Deltaphi) measurements. We find two (temperature-dependent) interaction/reaction channels: At 200 K, CO forms molecular (alpha) states with binding energies between 95 and 118 kJ/mol. At a coverage of Theta(CO) = 0.5, a c(2 x 2) LEED phase forms; for Theta(CO) > 0.5, a dim (2 x 3) structure is observed. The work function change of Deltaphi(max) = +780 meV indicates negatively polarized adsorbed CO molecules. Exposure at 500 K exclusively yields dissociated CO as deduced from high-temperature beta desorption states (185 kJ/mol < E-des < 210 kJ/mol), second-order reaction kinetics, and the formation of a sharp (1 x 2) LEED pattern at Theta(CO) = 0.25 (Theta(C) + Theta(O) = 0.50) associated with the least strongly bound beta(1) CO state. Deltaphi(max) amounts to merely +400 meV. Smaller CO coverages (0.125 < Theta < 0.16) give rise to a c(2 x 4) LEED structure. Upon heating, but prior to desorption, both superstructures undergo an order-disorder phase transition. Our results are discussed in terms of face specifities of CO dissociation, CO binding states, and structures and are compared with previous work on CO interaction/dissociation on related transition-metal surfaces.