After a brief introduction of Born's reciprocal relativity theory is presented, we review the construction of the deformed Quaplectic group that is given by the semi-direct product of U (1,3) with the deformed (noncommutative) Weyl-Heisenberg group corresponding to noncommutative fiber coordinates and momenta $ [ X_a, X_b ] ot= 0$; $[ P_a, P_b ] ot=0$. This construction leads to more general algebras given by a two-parameter family of deformations of the Quaplectic algebra, and to further algebraicextensions involving antisymmetric tensor coordinates and momenta of higher ranks $ [ X_{a_1 a_2 cdots a_n}, X_{b_1 b_2 cdots b_n} ] ot= 0$; $[ P_{a_1 a_2 cdots a_n}, P_{b_1 b_2 cdots b_n} ] ot=0$. We continue by examining algebraic extensions of the Yang algebra in extended noncommutative phase spaces and compare them with the above extensions of the deformed Quaplectic algebra. A solution is found for the exact analytical mapping of the non-commuting $ x^mu, p^mu$ operator variables (associated to an $8D$ curved phase space) to the canonical $ Y^A, Pi^A$ operator variables of a flat $12D$ phase space. We explore the geometrical implications of this mapping which provides, in the $classical$ limit, with the embedding functions $ Y^A (x,p), Pi^A (x,p) $ of an $8D$ curved phase space into a flat $12D$ phase space background. The latter embedding functions determine the functional forms of the base spacetime metric $ g_{mu u} (x,p) $, the fiber metric of the vertical space $h^{ab}(x,p)$, and the nonlinear connection $N_{a mu} (x,p) $ associated with the $8D$ cotangent space of the $4D$ spacetime. Consequently, one has found a direct link between noncommutative curved phase spaces in lower dimensions to commutative flat phase spaces in higher dimensions.