Based on symmetry principles, we derive a fusion algebra generated from repeated fusions of the irreducible modules appearing in the W-extended logarithmic minimal model WLM(p, p'). In addition to the irreducible modules themselves, closure of the commutative and associative fusion algebra requires the participation of a variety of reducible yet indecomposable modules. We conjecture that this fusion algebra is the same as the one obtained by application of the Nahm-Gaberdiel-Kausch algorithm and find that it reproduces the known such results for WLM( 1, p') and WLM(2, 3). For p > 1, this fusion algebra does not contain a unit. Requiring that the spectrum of modules is invariant under a natural notion of conjugation, however, introduces additional (p - 1)(p' - 1) reducible yet indecomposable rank-1 modules, among which the identity is found, still yielding a well-defined fusion algebra. In this greater fusion algebra, the aforementioned symmetries are generated by fusions with the three irreducible modules of conformal weights Δkp-1,1, k = 1, 2, 3. We also identify polynomial fusion rings associated with our fusion algebras.