[Atomic-S (OP)]

The radiative intensity of a blackbody depends only on temperature. The radiative intensity of a real body will in general be determined by temperature and also an efficiency factor that depends upon the material. Is there such a thing as a material that has different efficiency factors in different directions? Such a material, if formed into a sphere, would, if heated and then placed in a cold environment, radiate with different intensities in different directions.

[Soul Surfer]

Most definitely yes and you are very familiar with this. 

You are familiar that Shiny bright and white surfaces reflect the light energy falling them upon them.  They stay cooler when placed in the sun.  Dull dark and black surfaces absorb it and therefore get much hotter when placed in the sun.

This effect. by the laws of electromagnetics are reciprocal so a good reflector is a poor radiator and a poor reflector a good radiator.  You can think of this is the energy inside the body being reflected back into the body when it reaches the surface.

This is why an ideal radiator is described as a black body.

Now if you have something that has a shiny part and a dull part the radiations and absorptions are asymmetrical.

This effect is used to control the temperature inside space vehicles.

[Phractality]

Most definitely yes and you are very familiar with this. 

You are familiar that Shiny bright and white surfaces reflect the light energy falling them upon them.  They stay cooler when placed in the sun.  Dull dark and black surfaces absorb it and therefore get much hotter when placed in the sun.

This effect. by the laws of electromagnetics are reciprocal so a good reflector is a poor radiator and a poor reflector a good radiator.  You can think of this is the energy inside the body being reflected back into the body when it reaches the surface.

This is why an ideal radiator is described as a black body.

Now if you have something that has a shiny part and a dull part the radiations and absorptions are asymmetrical.

This effect is used to control the temperature inside space vehicles.

Sorry, Soul; your are misinformed. A black body is BOTH a good radiator AND a good absorber. The effect is directly proportional, not reciprocal.
I misread what Soul wrote. We are both saying the same thing.

Quoting Wikipedia, "
A black body in thermal equilibrium has two notable properties:^[1]

• It is an ideal emitter: it emits as much or more energy at every frequency than any other body at the same temperature.
• It is a diffuse emitter: the energy is radiated isotropically, independent of direction."

I don't know of any substance which radiates and absorbs nonisotropically. If there is such a thing, it doesn't fit the defintion of a black body. There are substances which have a different index of refraction for light polarized in different planes. Perhaps there are substance with different albedos in different directions, relative to their crystalline structure.
It is possible to combine a black body with a reflector to get nonisotropic absorption and a reflection. For example, you can place the black body at the focus of an elliptical reflector to absorb and emit selectively along the axis of the ellipse.
Retroreflective surfaces, reflect light back where it came from, rather than scattering it randomly. This is a property of the geometry of the reflector, rather than the substance. This can be accomplished with large glass beads or a paste made of microscopic glass beads. If the beads are the size of molecules, you might say that retroreflectance is a property of the substance.
Perpendicular corner reflectors do retroreflectance even better than glass beads.
If you place a latice of parallel reflecting planes in front of a balck body, you will get greater absorption if the light comes in parallel to the reflecting planes.
As we get better at making carbon nanotubes, we might be able to produce a sheet made of tightly packed parallel tubes, which reflect light in one direction and absorb in another direction.
You may have noticed shifting color bands as you view the surface of a CD or DVD. This results from tiny bumbs on the reflecting surface, which absorb specific wavelengths by interference. A similar effect produces the colors on a butterflies wing.
I would not be surprised if a butterflies wing is exactly what Atomic is looking for.

Under an electron microscope, you see bumps; in visible light, you see colors.

[Atomic-S (OP)]

It seems pretty clear on the basis of this discussion that materials can exist that have different coefficients of thermal radiation in different directions. Such materials are, indeed, not blackbodies, because to be a blackbody the efficiency of absorbtion or reflection must be 100% no matter what the direction, which anisotropic materials obviously are not. It appears that such materials can take various forms, one of which may be crystal anisotropy that accounts for the phenomenon of pleochroism, in which the color of the material looks different in different directions. Another obvious possible form is that of very thin layers of highly absorbtive material separated by transparent material. The cross section for radiation would be strongly dependent on direction.

[lightarrow]

The radiative intensity of a blackbody depends only on temperature. The radiative intensity of a real body will in general be determined by temperature and also an efficiency factor that depends upon the material. Is there such a thing as a material that has different efficiency factors in different directions? Such a material, if formed into a sphere, would, if heated and then placed in a cold environment, radiate with different intensities in different directions.

My pocket flashlight. 
Yes, the tungsten filament is not spherical, but I have no doubt that it worked the same even in that case.