Let φ: ℝ 2 → ℝ 2 be an orientation-preserving C 1 involution such that φ(0) = 0. Let Spc(φ) = {Eigenvalues of Dφ(p) {pipe} p ∈ ℝ 2}. We prove that if Spc(φ) ⊂ ℝ or Spc(φ) ∩ [1, 1 + ε) = ∅ for some ε > 0, then φ is globally C 1 conjugate to the linear involution Dφ(0) via the conjugacy h = (I + Dφ(0)φ)/2,where I: ℝ 2 → ℝ 2 is the identity map. Similarly, we prove that if φ is an orientation-reversing C 1 involution such that φ(0) = 0 and Trace (Dφ(0)Dφ(p) > - 1 for all p ∈ ℝ 2, then φ is globally C 1 conjugate to the linear involution Dφ(0) via the conjugacy h. Finally, we show that h may fail to be a global linearization of φ if the above conditions are not fulfilled. © 2012 Sociedade Brasileira de Matemática.434637653Alarcón, B., Guíñez, V., Gutierrez, C., Planar embeddings with a globally attracting fixed point (2008) Nonlinear Anal., 69 (1), pp. 140-150Alarcón, B., Gutierrez, C., Martínez-Alfaro, J., Planar maps whose second iterate has a unique fixed point (2008) J. Difference Equ. Appl., 14 (4), pp. 421-428Bialynicki-Birula, A., Rosenlicht, M., Injective morphisms of real algebraic varieties (1962) Proc. Amer. Math. Soc., 13, pp. 200-203Braun, F., dos Santos, F.J.R., The real Jacobian conjecture on ℝ 2is true when one of the components has degree 3 (2010) Discrete Contin. Dyn. Syst., 26 (1), pp. 75-87Cartan, H., Sur les groupes de transformations analytiques (1935) Actualités Scientifiques Et Industrielles, , Paris: HermannChamberland, M., Characterizing two-dimensional maps whose Jacobians have constant eigenvalues (2003) Canad. Math. Bull., 46 (3), pp. 323-331Chamberland, M., Meisters, G., A mountain pass to the Jacobian Conjecture (1998) Canad. Math. Bull., 41 (4), pp. 442-451Cima, A., Gasull, A., Mañosas, F., Global linearization of periodic difference equations (2012) Discrete Contin. Dyn. Syst., 32 (5), pp. 1575-1595Cima, A., Gasull, A., Mañosas, F., Simple examples of planar involutions with non-global Montgomery-Bochner linearizations (2012) Appl. Math. Lett., , doi: 10. 1016/j. aml. 2012. 05. 004Cobo, M., Gutierrez, C., Llibre, J., On the injectivity of C 1maps of the real plane (2002) Canad. J. Math., 54 (6), pp. 1187-1201Constantin, A., Kolev, B., The theorem of Kerékjártó on periodic homeomorphisms of the disc and the sphere (1994) Enseign. Math., 40 (3-4), pp. 193-204Fernandes, A., Gutierrez, C., Rabanal, R., On local diffeomorphisms of ℝ nthat are injective (2003) Qual. Theory Dyn. Syst., 4 (2), pp. 255-262Fernandes, A., Gutierrez, C., Rabanal, R., Global asymptotic stability for differentiable vector fields of ℝ n (2004) J. Differential Equations, 206 (2), pp. 470-482Feßler, R., A proof of the two-dimensional Markus-Yamabe stability conjecture and a generalization (1995) Ann. Polon. Math., 62 (1), pp. 45-74Gutierrez, C., A solution to the bidimensional global asymptotic stability conjecture (1995) Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (6), pp. 627-671Gutierrez, C., Jarque, X., Llibre, J., Teixeira, M.A., Global injectivity of C 1maps of the real plane, inseparable leaves and the Palais-Smale condition (2007) Canad. Math. Bull., 50 (3), pp. 377-389Gutierrez, C., Rabanal, R., Injectivity of differentiable maps ℝ 2 → ℝ 2at infinity (2006) Bull. Braz. Math. Soc., 37 (2), pp. 217-239Gutierrez, C., Pires, B., Rabanal, R., Asymptotic stability at infinity for differentiable vector fields of the plane (2006) J. Differential Equations, 231 (1), pp. 165-181Gutierrez, C., Sarmiento, A., Injectivity of C 1maps ℝ 2 → ℝ 2at infinity and planar vector fields (2003) Astérisque, 287, pp. 89-102Gutierrez, C., Teixeira, M.A., Asymptotic stability at infinity of planar vector fields (1995) Bull. Braz. Mat. Soc., 26 (1), pp. 57-66Hirsch, M.W., Fixed-point indices, homoclinic contacts, and dynamics of injective planar maps (2000) Michigan Math. J., 47 (1), pp. 101-108MacKay, R.S., Advanced Series in Nonlinear Dynamics (1993) Renormalisation in area-preserving maps, 6. , World Scientific Publishing CoMancini, S., Manoel, M., Teixeira, M.A., Divergent diagrams of folds and simultaneous conjugacy of involutions (2005) Discrete Contin. Dyn. Syst., 12 (4), pp. 657-674Markus, L., Yamabe, H., Global stability criteria for differential systems (1960) Osaka Math. J., 12, pp. 305-317Meyer, K.R., Hamiltonian systems with a discrete symmetry (1981) J. Differential Equations, 41 (2), pp. 228-238Montgomery, D., Zippin, L., (1955) Topological transformation groups, , Interscience PublishersOlech, C., On the global stability of an autonomous system on the plane (1993) Contributions to Differential Equations, 1, pp. 389-400Rabanal, R., Center type perfomance of differentiable vector fields in the plane (2009) Proc. Amer. Math. Soc., 137 (2), pp. 653-662Smyth, B., Xavier, F., Injectivity of local diffeomorphisms from nearly spectral conditions (1996) J. Differential Equations, 130 (2), pp. 406-414Teixeira, M.A., Local and simultaneous structural stability of certain diffeomorphisms (1981) Lecture Notes in Math., 898. , Springer