This work proposes a complementary extension of the factorial function to the real line, based on the separation of the bilateral integral of e^(-Ax) into two disjoint domains. For A>0, the classical Gamma function Γ(z) emerges in the right half-plane; for A<0, we introduce the Symmetric Gamma Function Γ ˉ(z)=e^(-iπ(z+1)) Γ(-z), defined in the left half-plane, with poles at the positive integers. The duality between the integral kernels underlies a regularization mechanism for logarithmic divergences: Γ_R (z)=1/Γ ˉ(z-1) and Γ ˉ_R (z)=1/Γ(z+1), replacing each pole by an exact finite value. An alternative real representation B(x)=∫_0^∞e^(-u) u^(-x) du (convergent for x<1) is constructed, together with a trigonometric factor C(x)=2sinu2061(πx)+cosu2061(πx), defining F(x)=C(x)B(x) for x≤0, yielding F(-n)=(-1)^n n!, thereby unifying the complex branch, real regularization, and Laurent expansion formalisms. The relation F(-n)⋅Resu2061(Γ,-n)=1 establishes the fundamental duality, which emerges directly from the construction of the integral kernels. The method is validated in arithmetic progressions, QED, and QCD. The extension to k-loops is systematic via exponentiation: Γ_R^((k) ) (-n)=[(-1)^n/n!]^k. Distinct physical prediction: In d=5 dimensions, the vacuum energy density changes sign — a repulsive force where the MS-bar predicts attraction. This inversion has direct implications for the stability of extra dimensions in string theory and brane models.