Given objects $x_1, \dotsc, x_n$, is there a name for the group generated by $x_1,\dotsc,x_n$ subject only to the relations $x_i^2 = 1$ for all $i \in \{1,\dotsc,n\}$?
The dihedral group seems similar (generated by reflections) but has more relations.
This group is known as the universal Coxeter group on $n$ generators. Every Coxeter group on $n$ generators is a quotient of this group. There is a corresponding Kac-Moody group, Lie algebra, generalized flag variety, etc. It has been known for some time that its Kazhdan-Lusztig polynomials have positive integer coefficients. A reference is Combinatorics of Coxeter groups by Bjorner and Brenti.