Summary: The symmetrization map \(\pi :\mathbb{C}^2 \rightarrow\mathbb{C}^2\) is defined by \(\pi (z_1, z_2)=(z_1 +z_2, z_1 z_2)\). The closed symmetrized bidisc \(\Gamma\) is the symmetrization of the closed unit bidisc \(\overline{\mathbb{D}^2}\), that is, \[ \Gamma = \pi (\overline{\mathbb{D}^2})=\{ (z_1 +z_2, z_1 z_2)\,:\, |z_i| \leq 1, i=1,2\}. \] A pair of commuting Hilbert space operators \((S, P)\) for which \(\Gamma\) is a spectral set is called a \(\Gamma\)-contraction. Unlike the scalars in \(\Gamma\), a \(\Gamma\)-contraction may not arise as a symmetrization of a pair of commuting contractions, even not as a symmetrization of a pair of commuting bounded operators. We characterize all \(\Gamma\)-contractions which are symmetrization of pairs of commuting contractions. We show by constructing a family of examples that even if a \(\Gamma\)-contraction \((S,P)=(T_1 +T_2, T_1 T_2)\) for a pair of commuting bounded operators \(T_1, T_2\), no real number less than 2 can be a bound for the set \(\{ \Vert T_1 \Vert, \Vert T_2 \Vert\}\) in general. Then we prove that every \(\Gamma\)-contraction \((S, P)\) is the restriction of a \(\Gamma\)-contraction \((\widetilde{S}, \widetilde{P})\) to a common reducing subspace of \(\widetilde{S}, \widetilde{P}\) and that \((\widetilde{S}, \widetilde{P})=(A_1 +A_2, A_1 A_2)\) for a pair of commuting operators \(A_1, A_2\) with \(\max \{\Vert A_1\Vert, \Vert A_2\Vert\} \leq 2\). We find new characterizations for the \(\Gamma\)-unitaries and describe the distinguished boundary of \(\Gamma\) in a different way. We also show some interplay between the fundamental operators of two \(\Gamma\)-contractions \((S, P)\) and \((S_1, P)\).