The paper presents new examples of high-rate bivariate bicycle (BB) codes with enhanced symmetry properties, which support fold-transversal Clifford gates without overhead. These codes feature explicit bases of logical operators and admit certain fold-transversal Clifford gates. The authors construct $[[98, 6, 12]]$ and $[[162, 8, 12]]$ BB codes that admit interesting fault-tolerant Clifford gates. The work also lays the mathematical foundations for explicit bases of logical operators and fold-transversal gates in quantum two-block and group algebra codes, which may be of independent interest. The paper discusses the homological algebra of CSS codes, the concept of purity and principal codes, and the construction of fold-transversal gates for BB codes. It provides detailed examples and analyzes the action of fold-transversal gates on logical operators.The paper presents new examples of high-rate bivariate bicycle (BB) codes with enhanced symmetry properties, which support fold-transversal Clifford gates without overhead. These codes feature explicit bases of logical operators and admit certain fold-transversal Clifford gates. The authors construct $[[98, 6, 12]]$ and $[[162, 8, 12]]$ BB codes that admit interesting fault-tolerant Clifford gates. The work also lays the mathematical foundations for explicit bases of logical operators and fold-transversal gates in quantum two-block and group algebra codes, which may be of independent interest. The paper discusses the homological algebra of CSS codes, the concept of purity and principal codes, and the construction of fold-transversal gates for BB codes. It provides detailed examples and analyzes the action of fold-transversal gates on logical operators.