We present a library of 139 near-IR spectra of cool asymptotic giant branch stars that will be useful for comparison with theoretical model atmosphere calculations and for modelling the integrated emission from intermediate-age stellar populations. The source list was selected from the 'extremely red' region of the Isaac Newton Telescope (INT) Photometric Hα Survey (IPHAS) colour–colour plane that is overwhelmingly dominated by very late-type stars. The spectral library also includes a large fraction of S-type and carbon stars. We present a number of spectral classification sequences highlighting the various molecular features identified and discuss a number of rare features with uncertain identifications in the literature. With its focus on particularly cool photospheres, this catalogue serves as a companion to recent spectroscopic atlases of MK standards in the near-IR. Finally, the relationship between IPHAS (r'−i') and (r'− Hα) colours and spectroscopically determined properties is discussed and a strong correlation between the (r'− Hα) colour and the C/O abundance index for S-type and carbon stars is noted. This relation has the potential to separate O-rich, S-type and carbon stars in the Galaxy based on their photometry alone.