We give a characterization of morphisms which preserve finite and infinite standard Sturmian words. The class of such morphisms coincides with the monoid {D,E}^* of the endomorphisms of A^*, where A = a,b, generated by the two elementary morphisms, E which interchanges the letter a with b and D which is the Fibonacci morphism defined as: D(a) = ab,D(b) = a. Some new properties of these morphisms are shown. In particular, we prove the following result: Let {X_n}_{n 0} be the sequence of sets inductively defined as: X₀ = {}, X₁ = {a}, X_{n + 1} = (AX_n)⁽⁻⁾ for n> 1, where denotes the empty word and (−) is the operator of left-palindrome closure associating to each word w the shortest word having w as suffix. For all n 0 one has: X_{n + 1} = ((E ι) G D)(X_n)a, where ι denotes the identity map. From this we derive a new characterization of the set PER of all words w having two periods p and q which are coprimes and such that ¦w¦ = p + q − 2. Indeed, one has that