The Dirac equation (DE) is a cornerstone of quantum physics. We prove that direct quantization of the 4-momentum vector p with modulus equal to the rest energy m (c = 0) yields a coordinate-free and manifestly covariant equation. In coordinate representation, this is equivalent to DE with spacetime frame vectors xu replacing Dirac’s gamma-matrices. Recall that standard DE is not manifestly covariant. Adding an independent Hermitian vector x5 to the spacetime basis a allows accommodating the momentum operator in a real vector space with a complex structure arising alone from vectors and multivectors. The real vector space generated from the action of the Clifford or geometric product onto the quintet {x0, x1, x2, x3, x5} has dimension 32, the same as the equivalent real dimension for the space of Dirac matrices. x5 proves defining for the combined CPT symmetry, distinction of axial vs. polar vectors, left and right handed rotors & spinors, etc. Therefore, we name it reflector and {x0, x1, x2, x3, x5} a basis for spacetime-reflection (STR). The pentavector I = x05123 commutes with all elements of STR and depicts the pseudoscalar in STR. We develop the formalism by deriving all essential results from the novel STR DE: spin ½ magnetic angular momentum, conserved probability currents, symmetries and nonrelativistic approximation. In simple terms, we demonstrate how Dirac matrices are a redundant representation of spacetime-reflection frame vectors.