Is there a substance that doesn't reflect OR absorb light from the visible light spectrum? Is there a substance that doesn't reflect or absorb visible light but may reflect light from another spectrum? Is there a theoretical substance that would have these properties?
EDIT:
Sorry I wasn't quite clear with my original question. I've updated it to express more what I was thinking. Would a substance be "invisible" if it didn't reflect or absorb light? Does a real or theoretical substance like that exist? I assume we can see glass and so on because it refracts light and to a certain extent reflects it.
 A: Yes, to the approximate extent allowed in the real world.
The condition that the material doesn't reflect visible light means that the material looks black. So consider various black coatings, for example, and ask what they do with electromagnetic waves at different frequencies than visible light.
Of course that you find out that they generally reflect much of this radiation, see e.g. the graph at the bottom

http://www.acktar.com/category/BlackOpticalCoating

to see that some of the commercial black coatings reflect infrared radiation (wavelength of several millimeters) much more than they reflect the visible light. I think that materials that are black in the visible range but reflect ultraviolet or more extreme radiation may also exist.
Update
The question was updated – whether a material may fail both to reflect and absorb visible light. No. Incoming energy must be either reflected or absorbed – by energy conservation. At most, one may have non-linear materials that are able to re-emit the incoming energy through different-frequency photons. But the energy can't get "lost".
A: It's surprisingly difficult to get a material that absorbs almost no light across the whole visible range but one candidate would be black silicon. This has a textured surface created by etching, and the texture means light hitting the surface is multiply reflected sideways before it gets a chance to be reflected back away from the surface. With some absorption at each reflection the multiple reflections mean the overall reflectivity can be a few percent. The material still reflects in the medium to far infra-red because the length scale of the texture is smaller than the wavelength of medium to far infra-red light.
An example of the relectance spectrum is given here:

Response to question v2:
Assuming I understand what you're getting at, the only technology I know of that would fit is metamaterials and in particular metamaterial cloaking. Metamaterial cloaks are effectively waveguides that can bend light around an object so the object in effect neither reflects nor absorbs light.
I should add that we are a long way from actually achieving this in any useful way, and so far what invisible materials do exist work only at microwave wavelengths. However the limitations are essentially technical and the idea would work if we could but fabricate the structures required.
A: Lol it's all around you. We call it air. It doesn't absorb or reflect light so it's usually invisible but it does reflect intense light as found in a laser beam. 
A: Materials can do more with light than just Reflect or Absorb.
Besides Refraction, Diffusion, and others, materials can also be "Translucent".
IE: they ignore light, such that light passes through them. True translucence would render a material invisible. Materials that are strikingly CLEAR still have a surface disturbance which causes a small reflection or refraction that is perceivable. That is why you can see the surface of Glass but not the interior layers.
As a verb (like "to Reflect" or "to Absorb")  I suppose it would be "to Transluce"   ?
I don't think that is a real word anymore (according to spellcheck...), but,  perhaps it should be. Primarily we use the word Transmit when talking about light being passed through Fiber-optic cables, or we say the light is "Guided" by the material, but these words are somewhat inaccurate when describing the real process.
Ultimately, I would be much happier using the word Transluce, but it does sound a little weird if you put it in a sentence in certain tenses.
"The cable transluces 90% of the photons without error"
"The light is translucing"  "... it was transluced"
but, hey, it only sounds weird until everyone is doing it, right?
