The largest prime p that can be the order of an automorphism of a 2-(35,17,8) design is p=17, and all 2-(35,17,8) designs with an automorphism of order 17 were classified by Tonchev. The symmetric 2-(35,17,8) designs with automorphisms of odd prime order $p<17$ were also classified. In this paper we give the classification of all symmetric 2-(35,17,8) designs that admit an automorphism of order $p=2$. It is shown that there are exactly $11,642,495$ nonisomorphic such designs. Furthermore, it is shown that the number of nonisomorphic 3-(36,18,8) designs which have at least one derived 2-$(35,17,8)$ design with an automorphism of order 2, is $1,015,225$.