Circulant (block circulant) operators are an important class of operators, which have wide applications in science and engineering calculations such as quantum computation, time-series analysis and compressing sensing, etc. The generation of block symmetric r-circulant operators (block symmetric r-skew-circulant operators) can be seen as symmetrical generation of circulant operators and adding parameters r or -r to entries below the para-diagonal elements. Based on the idea of order reduction, and using the properties of the diagonalization of block symmetric r-circulant operator matrices and unitarily invariant (weakly unitarily invariant) norm, the operator norms and Schatten p-norm inequalities and equality results of the block symmetric r-circulant operators and the block symmetric r-skew-circulant operators derived from the sub-block are given.