Not a separate class of shares. Almost always, founder shares refer to the magnitude of the grant and the voting-rights agreement attached to it. ("Stock class" is a very specific term.)
This serves my point not yours, the process is often very misunderstood by those being promised the equity, and it's often "converted" in ways that dilutes or outright takes away most of the value by the time the employees ever see financial gains from the equity.
This is super uncommon.
> stocks which are traded on the NYSE so nobody can cook the books without commiting securities fraud
The exact same fraud rules apply to private and public stock.