Given the real Clifford algebra of a quadratic space with a given signature, we define a new product in this structure such that it simulates the Clifford product of a quadratic space with another signature different from the original one. Among the possible applications of this new product, we use it in order to write the Minkowskian Dirac equation over the Euclidean spacetime and to define a new duality operation in terms of which one can find self-dual and anti-self-dual solutions of gauge fields over Minkowski spacetime analogous to the ones over Euclidean spacetime and without needing to complexify the original real algebra. © 2001 Plenum Publishing Corporation.401229241Baylis, W., (1996) Clifford (Geometric) Algebras with Applications in Physics, Mathematics and Engineering, , Birkhäuser, BostonChisholm, J.S.R., Common, A.K., (1986) Clifford Algebras and Their Applications in Mathematical Physics, , Reidel, DordrechtClifford, W.K., Applications of Grassmann's Extensive Algebra (1878) American Journal of Mathematics, 1, pp. 350-358Delanghe, R., Brackx, F., Serras, H., (1993) Clifford Algebras and Their Applications in Mathematical Physics, , Kluwer, DordrechtDietrich, V., Habetha, K., Jank, G., (1998) Clifford Algebras and Their Applications in Mathematical Physics, , Kluwer, DordrechtFigueiredo, V.L., De Oliveira, E.C., Rodrigues, W.A., Covariant, algebraic and operator spinors (1990) International Journal of Theoretical Physics, 29, pp. 371-395Gilbert, J.E., Murray, M.A.M., (1991) Clifford Algebras and Dirac Operators in Harmonic Analysis, , Cambridge University Press, CambridgeGlimm, J., Jaffe, A., (1987) Quantum Physics. A Functional Integral Point of View, , Springer-Verlag, New YorkHestenes, D., (1966) Space-Time Algebra, , Gordon and Breach, New YorkLounesto, P., Clifford algebras and Hestenes spinors (1993) Foundations of Physics, 23, pp. 1203-1237Lounesto, P., (1997) Clifford Algebras and Spinors, , Cambridge University Press, CambridgeMicali, A., Boudet, R., Helmstetter, J., (1992) Clifford Algebras and Their Applications in Mathematical Physics, , Kluwer, DordrechtParra, J.M., On Dirac and Dirac-Darwin-Hestenes equations (1992) Clifford Algebras and Their Applications in Mathematical Physics, pp. 463-477. , Artibano Micali, R. Boudet, and J. Helmstetter, Editors, Kluwer, DordrechtPezzaglia, W.M., Adams, J.J., (1997) Should Metric Signature Matter in Clifford Algebra Formulations of Physical Theories?, , e-Print Archive gr-qc/9704048Porteous, I., (1995) Clifford Algebras and the Classical Groups, , Cambridge University Press, CambridgeRiesz, M., (1993) Clifford Numbers and Spinors, , Kluwer, DordrechtRodrigues, W.A., De Souza, Q.A.G., Vaz, J., Lounesto, P., Dirac-Hestenes spinor fields on Riemann-Cartan manifolds (1996) International Journal of Theoretical Physics, 35, pp. 1849-1900Van Nieuwenhuizen, P., Waldron, A., On Euclidean spinors and Wick rotations (1996) Physics Letters B, 389, pp. 29-36