The Straight Dope ran an explanation of why nomads often wear black clothing - it absorbs heat better from the body. On the other hand, white clothing reflects sunlight better. Is it possible to get the best of both worlds and wear clothing that is black on the inside and white on the outside?
Since both absorption and radiation depend on color in the same way, this would decrease the energy that is radiated off into the surroundings. So if radiation were the only way to lose heat color wouldn't matter, and in the long term you'd have the same temperature as the incoming light.
But since nomad's don't live in vacuum there are other ways of heat transportation. What color is best depends on which forms of energy transportation dominate.
Since I guess that you lose more heat to other sources than radiation, while a large part of the incoming heat comes from light, making them white on the outside and black on the inside could make sense. But there are complicating factors such as wind, vaporization of sweat,...
The color of a surface doesn't reliably indicate the emissivity at non-visible wavelengths. The color in the visible spectrum is more of a side effect than anything. Most thermal radiation around body temperature or room temperature happens in the infrared region, not the visible, and that's not reliably indicated by visible color:
The visibly transparent glasses are opaque to the body's infrared emissions, while the visibly opaque trashbag is transparent to infrared. So one property has no relation to the other.
The emissions of the sun occur mostly in the visible region, which is why white clothing reflects solar energy and stays cool while black clothing absorbs solar energy and gets hot:
But your body's thermal radiation is in the infrared, so this rule doesn't apply to the inside of clothing (unless your body is hot enough to radiate visible light, but then you have bigger problems).
Your basic idea would work, though, if you found a material that reflects visible light while transmitting infrared light (but that material would probably have the same properties on the inside and outside, and thus be visibly white on the inside, too).
Since outside interacts with the radiation both ways, it would not help. But, there's something that would:
You need to have clothes which are Black (totally absorbing) in visible light and White in Thermal-infrared wavelengths (totally reflecting). This color is called Red (well, to some extent: if you shift the spectrum into visible terms).
Direct sunlight falling on people walking in the desert calls for white or light colors.
The coolest sun shades are white, also sun umbrellas. Coolest clothing, black versus white, is light colored. The reason is that in direct sunlight the energy of the sun, could be 1200 watts/metersquare all falls on the cloth. White reflects the visible light, which is most energy from the sun, so much less is absorbed to reach the shaded tables underneath .
As one can see the dominant dress is colored white. Black is the color of mourning and it might be that some women wear black even in the sun, and one will see light colored or brightly colored clothing occasionally due to the human tendency for decorating the self, but white is predominant. I checked on google and the statement in your link that Tuareg wear black is not correct. Most photos are white or light colors (similar arguments). Maybe they wear black for camouflage at night if on the war path?
Flowing robes also can be seen , no tight clothing, so that the air circulates and by convection cools the body as it perspires.
So the question becomes: would a black ( black in the infrared, totaly absorbing infrared material ) lining inside a white robe help in cooling the body by removing infrared radiation faster? My guess is yes, but cannot guess how much faster. The infrared radiated from the body would be better absorbed and transmitted to the outside but the main mechanism would still be convection.
If the temperature inside clothing is the same as outside, the wear will not do anything.
This is because white surface worse adsorbes light but also it worse emits infrared radiation. Conversely, dark surface better adsorbs, but also better emits.
In the case the outside temperature higher than inside, such wear will help to be cooler, but even better wear would be that one which has the both sides white. Similarly, if you want to keep heat in winter night, you would prefer the all-white wear because it better keeps the difference in the temperatures.
Conversely if you want your temperature to be as close to that of environment as possible, you should prefer all-black wear.