In discontinuous Galerkin FEM (as in continuous FEM) you begin by deriving the weak form of the PDE you are attempting to solve. The weak formulation guarantees exact conservation of the function you are trying to solve multiplied by a set of linearly independent functions (termed test functions).
The next step is to expand the solution you are solving for as a linear combination of another set of linearly independent functions (termed basis functions).
In typical DG method, the test and basis functions are the same i.e., they are derived from the same space. The only exception to this I have found are for PDEs with odd functions for which the same test and basis functions cannot be used.
Therefore, my question is why do people prefer using the same test and basis functions? Aside from it being the "traditional" method is there a mathematical reasoning?