The algebraic structure underlying dimensional analysis is commonly said to be a finitely generated Abelian group, whose generating set is the set of base units (e.g. length, time, mass, charge, and temperature). The problem with this is that there exist pairs of derived units that are not commensurable despite having the same decomposition, with the canonical example being work and torque: both decompose to $T^{−2}ML^{2}$ (aka force times distance) but it doesn't make sense to add them.
It occurs to me that the problem with this pair of units is conflating two kinds of multiplication. Work is more accurately described as the line integral of force over a path, which is a generalized dot product; torque is the cross product of force with the distance between the point where the force is applied and the axis of rotation. So we ought to say that work decomposes to $T^{-2}ML\cdot L$ whereas torque decomposes to $T^{-2}ML\times L$ and they are not the same.
The question is then:
Does conflation of different kinds of multiplication account for all cases where standard dimensional analysis gives the same decomposition for two (or more) different, incommensurable derived quantities?
What is the algebraic structure I'm using when I say things like "$T^{-2}ML\times L$", and what are its properties? It has three operators (scalar, dot, and cross product; or, if we prefer bivectors to pseudovectors, then wedge product instead of cross product) and we now have to keep track of whether the base units have scalar or vector nature. One of the operators is non-commutative, but distributivity and (if we use wedge product) associativity still apply. I don't know what this is called.