In this paper, we define a new variant of the knapsack problem, the Symmetric-weight Chain Precedence Knapsack problem (SCPKP). The (SCPKP) is the core structure of the Hydro Unit Commitment problem, the latter being a production scheduling problem relative to hydroelectric plants. The (SCPKP) is shown to be NP-hard. Polyhedral symmetries, featured by the (SCPKP), are introduced as a generalization of the classical symmetries applying only on the constraints without restricting the values of two symmetric solutions to be equal. A polyhedral study focuses on inequalities with 0-1 coefficients. Necessary facet defining conditions are described through a new structure, called pattern, encoding the polyhedral symmetries of the (SCPKP). A dedicated two-phase Branch & Cut scheme is defined to exploit the symmetries on the pattern inequalities. Experimental results demonstrate the efficiency of the proposed scheme in particular with respect to default CPLEX and a family of cover inequalities related to the Precedence Knapsack Problem.