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 modeling the integrated emission from intermediate-age stellar populations. The source list was selected from the `extremely red region of the 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 (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.
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