Index modulated orthogonal frequency division multiplexing (IM-OFDM) is a newly proposed technique, which achieves significantly improved error performance and peak-to-average power ratio (PAPR) in comparison with classical OFDM due to the activation of partial subcarriers. However, in the presence of inter-carrier interference (ICI), error detection of the subcarrier indices may easily occur, such that the performance of IM-OFDM is severely degraded and can be even worse than classical OFDM. To solve this problem, in this paper, we propose to tailor the ideas of classical ICI self-cancellation and two-path cancellation to IM-OFDM with two mapping methods, i.e., symmetric mapping and mirror mapping. Thanks to the joint design of the grouping and mapping methods, the proposed schemes inherit the IM-OFDM virtue of partially activated subcarriers but with reduced ICI. Monte Carlo Simulations validate that the proposed schemes significantly outperform conventional OFDM with ICI cancellation in additive white Gaussian noise (AWGN) channels with frequency deviation without sacrifice of the spectral efficiency and increase of the computational complexity.