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It seems it’s still an active debate and area of research, but the answer is more complex than wavelengths and emissivity. If you want to know whether black or white is cooler in the sun, it depends on: the breathability or knit, the amount of UV hitting the skin, the amount of skin contact with the fabric, wind speed, relative humidity, how the fabric wets and wicks moisture, and more. We could look at a black trash bag and say, well it’s transparent to IR, and it blocks the visible spectrum, therefore it’s a good shirt material to keep one cool. And obviously that would be wrong. In the same way it’s wrong to say: a white shirt feels less hot when you touch it, therefore it keeps the wearer cooler.
Yes, all of that is obviously true between shirts, the question is about shirt color, which is almost entirely down to the pigments used in fabrication. In which case it is entirely due to the absorptivity, emissivity, reflectance, and opacity, of the pigment.
This isn't an active area of debate, it's an entirely empirical question or a hard modeling problem per shirt manufacturer. All of this is very solved science, and has become "an engineering problem"
Well if protection from solar heat is the goal, it will be hard to beat the “chrome dome” or reflective parasol. Sometimes the ground reflects quite a bit of heat from below, like snow. Then I guess a shirt might out-perform a parasol.