A toric orbifold $X$ is a quotient of a moment-angle complex, and is characterized by a simple polytope $P$ and a characteristic function $\lambda$. A fundamental problem is to understand the interplay between the topology of $X$, the combinatorial data of the pair $(P,\lambda)$, and the algebraic structure of its cohomology $H^*(X)$.

In this talk, I will present my recent work investigating how the pair $(P,\lambda)$ determines the Steenrod operations on the cohomology of a 4-dimensional toric orbifold $X$. I will also discuss various applications of Steenrod operations, including computations of the stable homotopy type and gauge groups of $X$, a partial solution to the cohomological rigidity problem, and a combinatorial criterion for the existence of a spin structure in the smooth case.