I found a patent application for Chewing gum possessing tooth cleaning effect and a teeth cleaning method. Paragraph 130 addresses polishing agents:
Consequently, a polishing material can be any material that does not abrade dental enamel and dentine. Typical materials include silica gels and precipitates, aluminas, phosphates, and mixtures thereof. Specific examples include dicalcium orthophosphate dihydrate, calcium pyrophosphate, Bamboo, tricalcium phosphate, hydrated alumina, beta calcium pyrophosphate, calcium carbonate, sodium polymetaphosphate, sodium hexametaphosphate, Calgen, Giltex, Quadrafos, Hagan phosphate, micromet, calcium phosphate dibasic, calcium monohydrogen phosphate, dicalcium orthophosphate secondary calcium phosphate, carbonic acid calcium salt, cacti, calcichew, calcidia, citrical, aragonite, calcite, valerite, aluminum oxide, alumina, silicon dioxide, silica, silicic anhydride, and resinous abrasive materials such as particulate condensation products of urea and formaldehyde and others such as disclosed in U.S. Pat. No. 3,070,510. Mixtures of polishing agents can also be used.
I don't know all those ingredients. The ones that looked interesting are food grade silica gel and bamboo (powdered form).
I was surpised to see calcichew and citracal in there, I'm not familiar with their abrasiveness.
Also, whole (not ground) Chia seeds.