Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)This paper studies reversibility of subsemigroups acting on homogeneous spaces. The reversor set of a subsemigroup is defined and it is related to the invariant control sets for semigroups acting on certain homogeneous spaces. Let G be a connected noncompact semi-simple Lie group with finite center. Let L be a subgroup of G. Assume that S is a subsemigroup of G with intS not equal empty set. The main result characterizes the reversibility of the S-action on G/L in terms of the actions of S and L on the flag manifolds of G.843472486Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)FAPESP [07/06896-5]CNPq [303755/09-1]