A brief introduction of the Extended Relativity Theory in Clifford Spaces (C-space) paves the way to the explicit construction of the generalized relativistic transformations of the Clifford multivector-valued coordinates in C-spaces. The most general transformations furnish a full mixing of the grades of the multivector-valued coordinates. The transformations of the multivector-valued momenta follow leading to an invariant generalized mass M in C-spaces which differs from m. No longer the proper mass appearing in the relativistic dispersion relation E^2 - p^2 = m^2 remains invariant under the generalized relativistic transformations. It is argued how this finding might shed some light into the cosmological constant problem, dark energy, and dark matter. We finalize with some concluding remarks about extending these transformations to phase spaces and about Born reciprocal relativity. An appendix is included with the most general (anti) commutators of the Clifford algebra multivector generators.