The quantum notation used in the "traditional" .par formatted line-by-line lists for each molecule in the HITRAN database was described in detail in the supplementary material of the HITRAN2020 paper [1]. This information is reproduced in the following document and has been updated for additional molecules and isotopologues included in HITRAN2024 [2]. It will be maintained for future updates to the database:

In addition, supplementary XSAMS notation (with more detail) is available for each line if one requests a custom output format to include this information. Details of the XSAMS notation is available on the International Atomic Energy Agency (IAEA) website.
 

References

[1] I. E. Gordon, L. S. Rothman, R. J. Hargreaves, R. Hashemi, E. V. Karlovets, F. M. Skinner, et al., "The HITRAN2020 molecular spectroscopic database", J. Quant. Spectrosc. Radiat. Transfer 277, 107949 (2022). [link to article]

[2] I. E. Gordon, L. S. Rothman, R. J. Hargreaves, F. M. Gomez, T. Bertin, C. Hill, et al., "The HITRAN2024 molecular spectroscopic database", J. Quant. Spectrosc. Radiat. Transfer, in press (2026) [link to article]