Plane wave propagation in bianisotropic media has been addressed in the literature through several mathematical formalisms. Recently, using geometric algebra, the authors have used two different approaches to this problem: (i) a general analytical framework for reciprocal bianisotropic media; (ii) a new geometrical perspective for reciprocal anisotropic media. In fact, although anisotropy is a particular case of bianisotropy, anisotropy has special geometric characteristics that are well worth a specific approach: by considering aligned electric and magnetic anisotropies (i.e., when the eigenvectors of the permittivity and permeability functions are the same) we have introduced a new classification scheme for anisotropic media. In this communication we will show how our general analytical framework for bianisotropy, which explores the new algebraic techniques brought up by geometric algebra, interplays with the geometric insights provided by our specific approach to anisotropy. In fact, by studying a general anisotropic medium using the two aforementioned approaches, the general analysis of bianisotropic media reveals new geometrical insights within the context of anisotropy.