By using the symmetric serial decomposition of a normalized Mueller matrix M [J. Opt. Soc. Am. A 26, 1109 (2009)] as a starting point and by considering the reciprocity property of Mueller matrices, the geometrical features of the Poincaré sphere mapping by M are analyzed in order to obtain a new parameterization of M in which the 15 representative parameters have straightforward geometrical interpretations. This approach provides a new geometry-based framework, whereby any normalized Mueller matrix M is completely described by a set of three associated ellipsoids whose geometrical and topological properties are characteristic of M. The mapping analysis considers the cases of type-I and type-II, as well as singular and nonsingular Mueller matrices. The novel parameterization is applied to several illustrative examples of experimental Mueller matrices taken from the literature.