  Let $C_2$ be the cyclic group of order two. We present a structure theorem
for the $RO(C_2)$-graded Bredon cohomology of $C_2$-spaces using coefficients
in the constant Mackey functor $\underline{\mathbb{F}_2}.$ We show that, as a
module over the cohomology of the point, the $RO(C_2)$-graded cohomology of a
finite $C_2$-CW complex decomposes as a direct sum of two basic pieces: shifted
copies of the cohomology of a point and shifted copies of the cohomologies of
spheres with the antipodal action. The shifts are by elements of $RO(C_2)$
corresponding to actual (i.e. non-virtual) $C_2$-representations. This
decomposition lifts to a splitting of genuine $C_2$-spectra.
