Not-so-hot black shirt As it is getting warmer here by the minute I was asking myself:
Are there materials, that are black (in th visible range) but reflect (most) invisible light?
Furthermore, I asked myself what fraction of heat could be kept away using this material in daylight.
For bonus points, it would be great if it could be used as some kind of fabric. I am not looking for a specific material but am curious, as e.g. metals do reflect all light up to a certain frequency and thus would not suffy. Therefore it would be nice to understand the underlying mechanism.
In the winter of course it would be nice to reverse this, so to have white shirts, still getting hot in the sun.
 A: As only about 50 % of the energy of the light is in the visible spectrum is in the visible area (Wikipedia) this should be at least theoretically possible.
So you would get with a close to perfect t-shirt only 50% of the energy.
Whether such materials do exist, I don't know.
I guess there are some materials fulfilling the condition, but I don't know whether they are cheap enough and flexible / light enough to be used as clothes.
A: There is apparently a whole industry dedicated to this (relevant google search near infrared reflective coating
For instance, see this paper in Thermal Performance: Special Infrared Reflective Pigments Make a 
Dark Roof Reflect Almost Like a White Roof by Bill Miller et al. 
As user3384414 pointed out, nearly half of the solar radiance is outside the visible, e.g. (from wikimedia commons)

We see that the near IR is much more important than trying to block the UV.
From the linked paper: 

It seems that for black surfaces, these kinds of coatings can change the solar reflectance from around 4% to 40% or so, even higher than the untreated reflectance of the gray surface (20%).
