Baxter numbers are known to count several families of combinatorial objects, all of which come equipped with a natural involution. In this paper, we add a combinatorial family to the list, and show that the known bijections between these objects respect these involutions. We also give a formula for the number of objects fixed under this involution, showing that it is an instance of Stembridge's ``$q=-1$ phenomenon''. A polynomial $\sum_{i=0}^{n} a_i t^i$ with symmetric coefficients ($a_{n-i}=a_i$) has a unique expansion $\sum_{k=0}^{\lfloor n/2 \rfloor} \gamma_k t^k(1+t)^{n-2k}$, and is said to be \emph{gamma-nonnegative} if $\gamma_k\geq 0$ for all $k$. We either prove or conjecture a stronger $q$-analogue of this property for several polynomials in two variables $t$,$q$, whose $q=1$ specializations are known to be gamma-nonnegative.

The overarching theme of this thesis is to study monoid algebras and cohomology rings of configuration spaces through the lens of symmetry, with the goal of answering questions, forging connections and translating methods between algebraic combinatorics, representation theory and algebraic topology. Chapters 3 and 4 of the thesis establish novel connections between cohomology rings of configuration spaces and classical combinatorial algebras arising in the theory of reflection groups and hyperplane arrangements. These chapters introduce new tools to describe the symmetries of several combinatorially significant topological spaces. Chapter 5 studies certain monoids called left regular bands from the perspective of invariant theory, developing a framework to study the free left regular band and its q-analogue in parallel.