Clifford (geometric) algebra sheds new light on linear algebra by prompting a fresh coordinate-free approach,
thereby providing a geometric alternative to the classical methods of tensor (or dyadic) analysis. In this communication,
we develop a general framework for reciprocal bianisotropic media through this fresh perspective. As a
result, the interplay between anisotropy and magnetoelectric coupling emerges. Namely, the eigenwaves associated
with the (electric and magnetic) anisotropy and bianisotropy (i.e., the magnetoelectric coupling) are fully
characterized through the functional invariants which receive a complete geometric interpretation.