We present a periodic density-functional investigation of the adsorption and the vibrational spectroscopy of CO in mordenite. Our results highlight a pronounced sensitivity of the strength of the hydrogen bond between the acidic hydroxyl groups and the adsorbed molecule, and hence of the induced red shift of the OH, and the blue shift of the CO stretching mode on the choice of the exchange-correlation functional. The popular Perdew-Wang (PW) gradient-corrected functional strongly overestimates the frequency shifts and the interaction energies. We demonstrate that the revised Perdew-Burke-Ernzerhof (RPBE) functional leads to an improved description of hydrogen bonding. For bridging OH groups, terminal silanol groups and for Lewis sites formed by tricoordinated Al atoms, we predict adsorption energies and frequency shifts in good agreement with experiment. The calculated difference in the binding energies of CO in purely siliceous mordenite and at Brønsted acid sites in the main channel agrees very well with microcalorimetry data. We find that Brønsted acid sites in the small channels (the side-pockets) of mordenite do not adsorb CO, which is adsorbed only in the main channel via the C atom. For these adsorption complexes we find reasonable (though not perfect) agreement of the predicted blue shift of the CO-mode and of the red shift of the OH-mode with experiment. Our prediction that the side-pockets are inaccessible to CO correlates well with the microcalorimetric studies and with experimental observation of the adsorption of O(2), N(2) and H(2) molecules but contradicts the current interpretation of experimental adsorption studies for CO.