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

A structure theorem for $RO(C_2)$-graded Bredon cohomology