This paper treats extension and retraction properties in the category *$/9 of compact metric spaces with periodic maps of a prime period p; the subspaces and maps in J^p are called equivariant subspaces and maps, respectively. The motivation of the paper is the following question: Let E be a Euclidean space and α: E X E-> E X E be the involution (x, y) -> (y, x), i.e., the symmetry with respect to the diagonal. Suppose that Z is a symmetric (i.e., equivariant) closed subset of ExE which is an absolute retract; that is, Z is a retract of E X E. When does there exist a symmetric (i.e., equivariant) retraction Ex E-+ZΊ This is an extension problem in the category J2/'p. If X and Y are spaces in J£fp, A is a closed equivariant subspace of X and /: A -> Y is an equivariant map, then the existence of an extension of / does not, in general, imply the existence of an equivariant extension. It is shown, however, that if A contains all the fixed points of the periodic map and dim(X— A) < oo, then a condition for the existence of an extension is also sufficient for the existence of an equivariant extension. In particular, it follows that a finite dimensional space X in Sf 'p is an equivariant ANR (i.e., an absolute neighborhood retract in the category Sf v) if and only if it is an ANR and the fixed point set of the periodic map on X is an ANR. Generally speaking, the paper deals with the question of symmetry in extension and retraction problems.