We have started an ambitious program to determine if the full diversity of extinction laws is real or if some of it is due to calibration or methodological issues. Here we start by analyzing the information on NIR extinction in a 2MASS stellar sample with good quality photometry and very red colours. We calculate the extinction at 1 μm, A1, and the power-law exponent, α (Aλ=A1λ−α), for the 2MASS stars located in the extinction trajectory in the H−K vs. J−H plane expected for red giants with A1>5 mag. We test the validity of the assumption about the nature of those stars, whether a single or multiple values of α are needed, and the spatial variations of the results. Most (∼83%) of those stars can indeed be explained by high-extinction red giants and the rest is composed of extinguished AGB stars (mostly O-rich), blended sources, and smaller numbers of other objects, a contaminant fraction that can be reduced with the help of Gaia} DR2 data. Galactic red giants experience a NIR extinction with α∼2.27 and an uncertainty of a few hundredths of a magnitude. There is no significant spread in α even though our sample is widely distributed and has a broad range of extinctions. Differences with previous results are ascribed to the treatment of non-linear photometric effects and/or the contaminant correction. Future research should concentrate in finding the correct functional form for the NIR extinction law. In the appendix we detail the treatment of non-linear photometric effects in the 2MASS bands.