We study a class of aggregate-join queries with multiple aggregation operators evaluated over annotated relations. We show that straightforward extensions of standard multiway join algorithms and generalized hypertree decompositions (GHDs) provide best-known runtime guarantees. In contrast, prior work uses bespoke algorithms and data structures and does not match these guarantees. Our extensions to the standard techniques are a pair of simple tests that (1) determine if two orderings of aggregation operators are equivalent and (2) determine if a GHD is compatible with a given ordering. These tests provide a means to find an optimal GHD that, when provided to standard join algorithms, will correctly answer a given aggregate-join query. The second class of our contributions is a pair of complete characterizations of (1) the set of orderings equivalent to a given ordering and (2) the set of GHDs compatible with some equivalent ordering. We show by example that previous approaches are incomplete. The key technical consequence of our characterizations is a decomposition of a compatible GHD into a set of (smaller) {\em unconstrained} GHDs, i.e. into a set of GHDs of sub-queries without aggregations. Since this decomposition is comprised of unconstrained GHDs, we are able to connect to the wide literature on GHDs for join query processing, thereby obtaining improved runtime bounds, MapReduce variants, and an efficient method to find approximately optimal GHDs.