We deal with germs of diffeomorphisms that are reversible under an involution. We establish that this condition implies that, in general, both the family of reversing symmetries and the group of symmetries are not finite, in contrast with continuous-time dynamics, where typically there are finitely many reversing symmetries. From this we obtain two chains of fixed-points subspaces of involutory reversing symmetries that we use to obtain geometric information on the discrete dynamics generated by a given diffeomorphism. The results are illustrated by the generic case in arbitrary dimension, when the diffeomorphism is the composition of transversal linear involutions.
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