This preprint develops a signed-XOR structural-constant calculus for basis multiplication in Clifford and Cayley–Dickson type algebras. The main idea is to separate the common XOR support law from the algebra-specific sign twist. In the sorted Clifford sector, the paper proves a unique normal-form theorem, derives a closed signed-XOR product formula, identifies the finite case with real Clifford algebras, and expresses the structural constants in bit-vector form. For general signed-XOR twists, it computes associators, commutators, squares, and gauge transformations, and recovers Cayley–Dickson basis multiplication through a recursive twist. The work is intended as a table-free structural-constant framework for computational algebra. Its numerical claim is conservative: sparse multiplication retains the usual bilinear dependence on stored terms, but each encountered basis product can be evaluated by XOR and parity without storing a full basis multiplication table.