Mineral Spirits is OK for Ivory,
You will see it listed as 'White Spirits' in some museum conservation process methods, especially of British origin.
But use it as a surface cleaner by dampening a cloth pad or use a cotton swab dampened with it. Then clean the surface off afterwards with a mild soap & water useing the same method & dry it.
It's great for getting tape and sticker glue and gum off of ivory surfaces. Some marker and other writing ink can also be removed with it. But as the ivory is porous, sometimes the marker matr'l has been absorbed deeper down into the ivory surface and will still leave a ghost image.
Never dunk/soak ivory in anything including water.
Lots of people rub a coating of Mineral Oil on ivory objects. They feel it 'feeds' the surface. Ivory is porous like wood is and absorbs water and other things in the same manner.
Mineral oil/Baby Oil,,it's all the same stuff. Just highly refined petro oil to the edible stage .
A very very light coating rubbed on, then excess removed with a soft cloth won't do any harm.
I'd avoid any drying oils like wood finish type and even Linseed Oils which are gummy and sometimes never really dry but can be absorbed by the Ivory. Linseed has a habit of turning dark as it oxidizes, even near black over time.
Keep Ivory from extremes in temperature and humidity. High temp and low humidity will cause cracking. Sometimes within a few hours if really excessive like around a wood stove for example.
The cracks won't heal themselves by simply re-introducing the ivory piece to higher humidity and lower temps once again. The gaps may close up, but the separation in the ivory is still there. Then the dirt and grime from simple handling gets in there and is nearly impossible to get out.
Pearl(s) and Mother of Pearl are the same substance.
Very fragile, much more than Ivory.
What you can use to clean Ivory is pretty much off limits on Pearls and MOP if you want to keep them in top shape.
No acidic cleaners, no abrasives even mild ones, no Mineral Spirits, Acetone or the like.
A warm water and very mild soap damp wipe down, then the same room temp water damp rinse and dry is the recommendation for jewelry MOP pieces.
The high polish color layer on top can be damaged with things as simple as personal hygene products, sweat, etc.
But talking handgun grips, they are going to get handled much more heavily than jewelry, so the orig just made new look will not remain long unless they are white glove handled and safe queen stored.
Just keep them as clean as possible with the above processes and realize
they are fragile and can crack and chip easily as well as be surface damaged from normal gun handling procedures.
It's just the nature of the natural material they happen to be made of.
Either of the materials will resist cracking or other damage in use if they are closely fitted to the handgun frame and are securely held in position with no gaps to the frame.
Ivory grips are usually much better fit than the average MOP set of grips. The latter generally don't have the exacting , tight fit around the edges of the grip frame nor the excellent flat fit to the frame to keep them from 'rocking'. Keep the grip screw(s) tight, but don't over tighten them especially with the MOP grips.
There's always Mother of Plastic as an alternative...