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 H{alpha} 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'-H{alpha}) colours and spectroscopically determined properties is discussed and a strong correlation between the (r'-H{alpha}) 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.