Since the set of spin structures on a principal $SO(n)$-bundle on a manifold $X$ is in one to one correspondence with $H^1(X,Z_2)$, this group admits an action on spin structures. I wanted to know if we can interpret this action geometrically.
How about the same question for for action of $H^2(X,Z)$ on $spin^c$ structures on $X$?