Using a sample of 102 X 10(6) Y(1S) events collected with the Belle detector, we report on the first search for charge-parity-even charmonium and charmoniumlike states in Y(1S) radiative decays. No significant chi(cJ) or eta(c) signal is observed and 90% C. L. limits on B(Y(1S) -> gamma chi(c0)) < 6.5 X 10(-4), B(Y(1S) -> gamma chi(c1)) < 2.3 X 10(-5), B(Y(1S) -> gamma chi(c2)) < 7.6 X 10(-6), and B(Y(1S) -> gamma eta(c)) < 5.7 X 10(-5) are obtained. The product branching fraction limits B(Y(1S) -> gamma X(3872))B(X(3872) -> pi(+) pi(-) J/psi) < 1.6 X 10(-6), B(Y(1S) -> gamma X(3872))B(X(3872) -> pi+ pi(-) pi(0) J/psi) < 2.8 X 10(-6), B(Y(1S) -> gamma X(3915)) B(X(3915) -> omega J/psi) < 3.0 X 10(-6), and B(Y(1S) -> gamma Y(4140))B(Y(4140) -> phi J/psi) < 2.2 X 10(-6) are obtained at the 90% C.L. Furthermore, no evidence is found for excited charmonium states below 4.8 GeV/c(2).