As mentioned, the Altamount grips are made from laminated wood, often called (dymondwood). The wood is laminated by impregnating it under high pressure and heat with a Phenolic resin. It penetrates into the wood cells/fiber and become part of the wood structure.
There are variations of laminated wood... Boyd's Gunstock uses a laminated wood material called Stratabond... and as the name implies, Fender guitar uses the material in neck and bodies and Martin guitar used it for necks.
What results is something that can be machined and worked like wood, but behaves like plastic when it comes to being completely resistant to absorbing anything.
Once that resin sets up, that is it, the grips are completely sealed. Putting any kind oil or cleaner wax just lays on the surface.
The one thing really nice about laminated grips and rifle stocks like that is when they get scratched, you can sand out the scratch, then smooth with progressively fine grit sandpaper to around 800 grit, and then they can be buffed them back to the original condition (if shinny) using a cloth and polishing paste. Otherwise, after polishing you lightly burnish it with 0000 steel wool and rub with a cotton cloth to give it a satin sheen.
Because it's what I have on hand, I use what I use on the clear coat of my car to remove scratches... Meguire's ScratchX® 2.0. You could even use Flitz sparingly too.
It may look like wood, but think plastic as far as care and maintenance is concerned.