Using a sample of 158 x 10(6) gamma (2S) events collected with the Belle detector, charmonium and charmoniumlike states with even charge parity are searched for in gamma (2S) radiative decays. No significant chi(cJ) or eta(c) signal is observed, and the following upper limits at 90% confidence level (C. L.) are obtained: B(gamma(2S)-> gamma chi(c0)) < 1.0 x 10(-4), B(gamma(2S) -> gamma chi(c1)) < 3.6 x 10(-6), B(gamma(2S) -> gamma chi(c2)) < 1.5 x 10(-5), and B(gamma(2S) ->gamma eta(c)) < 2.7 x 10(-5). No significant signal of any charmoniumlike state is observed, and we obtain the limits B(gamma(2S) -> gamma X(3872)) x B(X(3872) -> pi(+)pi(-) J/psi) < 0.8 x 10(-6), B(gamma(2S) -> gamma X(3872)) x B(X(3872) -> pi(+)pi(-)pi(0) J/psi) < 2.4 x 10(-6), B(gamma(2S) -> gamma X(3915)) x B(X(3915) -> omega J/psi) < 2.8 x 10(-6), B(gamma(2S) -> gamma Y(4140)) x B(Y(4140) -> phi J/psi) < 1.2 x 10(-6), and B(gamma(2S) -> gamma X(4350) x B(X(4350) -> phi J/psi)) < 1.3 x 10(-6) at 90% C. L.