Eight Symmetry-Aware Rules for Symbolic Integration and Order Selection: A Group-Theoretic and Algorithmic View with Theorem-Example Format

We present eight high-level rules for symbolic integration and the selection of integration order, distilled from a group-theoretic and algorithmic perspective. Each rule is stated succinctly, then supported by theorem/proposition/lemma statements and worked examples. The contributions are organizational and computational: they augment classical results (e.g., Fubini-Tonelli, change of variables) with symmetry partitioning, projector-based cancellations, posetconstrained searches, cost-guided order selection, decorrelation, stabilizer-informed budgeting, blockwise structuring, and symmetry-driven variance reduction.