Let 12H(n,2) denote the halved n-cube with vertex set X and let T≔T(x0) denote the Terwilliger algebra of 12H(n,2) with respect to a fixed vertex x0∈X. In this paper, we assume n≥6. We first characterize T by considering the action of the automorphism group of 12H(n,2) on the set X×X×X. We show that T coincides with the centralizer algebra of the stabilizer of x0 in the automorphism group, and display three subalgebras of T further. Then we study the homogeneous components of V≔ℂX, each of which is a nonzero subspace of V spanned by the irreducible T-modules that are isomorphic. We give a computable basis for any homogeneous component of V. Finally, we describe the decomposition of T via its block-diagonalization and give a basis for the center of T by using the above homogeneous components of V.